ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
B. CONRAD, K. CONRAD, AND H. HELFGOTT
Abstract. For a global field K and an elliptic curve Eη over K(T ), Silverman’s specialization theorem implies rank(Eη (K(T ))) ≤ rank(Et (K)) for all but finitely many t ∈ P1 (K).
If this inequality is strict for all but finitely many t, the elliptic curve Eη is said to have elevated rank. All known examples of elevated rank for K = Q rest on the parity conjecture
for elliptic curves over Q, and the examples are all isotrivial.
Some additional standard conjectures over Q imply that there does not exist a nonisotrivial elliptic curve over Q(T ) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational
function field K = κ(u) over any finite field κ with characteristic 6= 2, we construct an
explicit 2-parameter family Ec,d of non-isotrivial elliptic curves over K(T ) (depending on
arbitrary c, d ∈ κ× ) such that, under the parity conjecture, each Ec,d has elevated rank.
To Mike Artin on his 70th birthday
1. Introduction
Let K be a global field and let Eη be an elliptic curve over K(T ). This curve uniquely
extends to a minimal regular proper elliptic fibration E → P1K . The group Eη (K(T )) is
finitely generated, by the Lang–Néron theorem [17, Thm. 1]. (See [5, §6] for a proof of the
Lang–Néron theorem using the language of schemes.) For all but finitely many t ∈ P1 (K),
the specialization Et of E at T = t is an elliptic curve over K. This paper is concerned with
a comparison between the ranks of Eη (K(T )) and Et (K) as t varies.
By Silverman’s specialization theorem [33, Thm. C], the specialization map
Eη (K(T )) = E (P1K ) → Et (K)
at T = t is injective for all but finitely many t ∈ P1 (K). (To be precise, Silverman’s
theorem only applies to non-constant Eη . Injectivity of the specialization map for constant
Eη is elementary.) Thus, the generic rank r(Eη ) := rank(Eη (K(T ))) satisfies
(1.1)
r(Eη ) ≤ rank(Et (K))
for all but finitely many t. The elliptic curve Eη (or the fibration E → P1K , or the family
{Et }t∈P1 (K) ) is said to have elevated rank if (1.1) is a strict inequality for all but finitely
many t ∈ P1 (K).
How are examples of elevated rank constructed? The only known technique depends on
the parity conjecture: for every elliptic curve E over the global field K,
?
(−1)rank(E(K)) = W (E),
2000 Mathematics Subject Classification. 11G05,11G40.
Key words and phrases. elliptic curve, root number, function fields.
We thank N. Boston, I. Dolgachev, A.J. de Jong, B. Mazur, D. Pollack, B. Poonen, K. Rubin, A.
Silberstein, E. Spiegel, R. Vakil, A. Venkatesh, and S. Wong for helpful conversations. B.C. thanks the NSF
and the Sloan Foundation for support.
1
2
B. CONRAD, K. CONRAD, AND H. HELFGOTT
where W (E) is the global root number of E. The spirit of the parity conjecture is that
W (E) is supposed to be the sign in the functional equation of the L-function of E, but
such a functional equation is not yet known to exist in general. Therefore, we adopt the
convention that the global root number is defined to be the product of local root numbers.
The local root numbers are defined in all cases via representation theory [6] and are equal
to 1 at non-archimedean places of good reduction. Some convenient formulas for local root
numbers at non-archimedean places will be recalled in Theorem 3.1. The analytic and
representation-theoretic descriptions of W (E) are known to agree when K is Q or a global
function field, by work of Deligne, Drinfeld, Wiles, and others. In particular, since our focus
in this paper will be the cases when K = Q or when K is a rational function field over a
finite field, the reader can think about W (E) in either way.
To find elliptic curves with elevated rank, one tries to construct E → P1K such that W (Et )
has opposite sign to (−1)r(Eη ) with at most finitely many exceptions. That is, we want
(1.2)
W (Et ) = −(−1)r(Eη )
for all but finitely many t ∈ P1 (K). Assuming the parity conjecture for elliptic curves over
K, (1.1) and (1.2) imply that (1.1) is a strict inequality for all but finitely many t, so the
Et ’s have elevated rank.
Because of the role of the parity conjecture in this strategy, all known examples of elevated
rank are, strictly speaking, conditional. Moreover, this idea has so far only been carried
out when K = Q. The first (conditional) examples of elevated rank were found by Cassels
and Schinzel [1]. These are quadratic twists over Q(T ) of the elliptic curve y 2 = x3 − x:
(1.3)
En,η : n(1 + T 4 )y 2 = x3 − x,
where n is a (squarefree) positive integer satisfying n ≡ 7 mod 8. Each En,η should have
elevated rank, because the group En,η (Q(T )) has rank 0 and W (En,t ) = −1 for every t ∈ Q.
We exclude t = ∞ because En,∞ is not smooth. (Although En,t (Q) should have a point of
infinite order for each t ∈ Q, there cannot be an algebraic formula for a point of infinite
order on En,t (Q) as t varies through an infinite subset of Q, since such a formula would give
an element of infinite order in the group En,η (Q(T )).) More generally, for any elliptic curve
E/Q , Rohrlich [28, Prop. 9] proved that there is a quadratic twist Eη of E/Q(T ) by a quartic
irreducible in Q[T ] such that Eη (Q(T )) has rank 0 and W (Et ) = −1 for every t ∈ Q.
Nekovář has proved the parity conjecture for any elliptic curve over Q with finite Tate–
Shafarevich group [23], [32, p. 463], but this does not make any examples of elevated rank
over Q unconditional, since there are no non-constant families E → P1Q such that Et is
known to have a finite Tate–Shafarevich group for all but finitely many t ∈ P1 (Q). Similarly,
the recent work of Kato and Trihan [15] (as well as earlier work of Artin–Tate, Milne,
Schneider, and others) on the Birch and Swinnerton-Dyer conjecture in characteristic p does
not have any impact on the conditional nature of the parity conjecture in characteristic p
as it is applied to the examples considered in this paper.
The examples of Cassels–Schinzel and Rohrlich over Q(T ) are quadratic twists. The
appeal of quadratic twists is that there are simple formulas that describe the variation
of root numbers under quadratic twists over Q [30, Cor. to Prop. 10], [32, Thm. 7.2].
However, a family of quadratic twists exhibits no “geometric” variation: it is isotrivial
(that is, j(Eη ) ∈ K, with no T -dependence), and conversely any isotrivial family is either
a family of quadratic twists or is a family of quartic (resp. cubic or sextic) twists with
j(Eη ) = 1728 (resp. j(Eη ) = 0).
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
3
The main question we address in this paper is the following: for a global field K, does
there exist a non-isotrivial elliptic curve over K(T ) with elevated rank? In Appendix A,
we explain why some standard conjectures over Q imply that the answer to this question
for K = Q is no. There are natural analogues of these standard conjectures over a rational
function field κ(u) over a finite field κ, but (as we explain in Appendix B) one of these
conjectures is false over κ(u). This suggests that our question might have an affirmative
answer in the function field case.
Here is our example. Let κ be a finite field with characteristic p 6= 2, and choose any
c, d ∈ κ× . Let F = κ(u) and consider the elliptic curve
(1.4)
Eη : y 2 = x3 + (c(T 2 + u)2p + du)x2 − (c(T 2 + u)2p + du)3 x
over F (T ). The Weierstrass model (1.4) over F (T ) has the form y 2 = x3 + Ax2 − A3 x. The
j-invariant j(Eη ) ∈ F (T ) is not in F , so Eη/F (T ) is non-isotrivial. An inspection of the poles
of j(Eη ) on P1F shows that changing (c, d) changes j(Eη ).
Let E → P1F be the minimal regular proper elliptic fibration with generic fiber Eη . For
all t ∈ P1 (F ), the specialization Et of E at T = t is an elliptic curve.
Theorem 1.1. Let F = κ(u), where char(κ) 6= 2, and fix a choice of c, d ∈ κ× . Let Eη/F (T )
be as in (1.4), depending on the choice of c and d. For every t ∈ P1 (F ), we have W (Et ) = 1.
If t 6= ∞, then Et (F ) has positive rank. Moreover, rank(Eη (F (T ))) = 1 and
W (Et ) = −(−1)rank(Eη (F (T )))
for all t ∈ P1 (F ).
Thus, if the parity conjecture is true for elliptic curves over F , the Et ’s are a non-isotrivial
family with elevated rank.
Remark 1.2. When t = ∞, the fiber Et in Theorem 1.1 is the constant elliptic curve
y 2 = x3 − c3 x over F . The elliptic curve E∞ therefore has global root number 1, and it
must have rank 0 (since F is a function field of genus 0 over the finite field κ).
We expect that Et (F ) has rank 2 except for a set of t ∈ P1 (F ) with density 0 (as measured
by height), but we have no idea how to prove this expectation.
Remark 1.3. We did not search for non-isotrivial examples of elevated rank with generic
rank 0 or in characteristic 2, but we expect that such examples exist. (The curve defined
by (1.4) in characteristic 2 is not smooth.) Our example has j(Eη ) ∈ F (T p ), and we expect
any example of elevated rank in characteristic p will have the same property.
Remark 1.4. For fixed p 6= 2, consider the algebraic family Et where κ and (c, d) vary (in
characteristic p) but the logarithmic height of t ∈ κ(u)× (i.e., the maximum of the degrees
of its numerator and denominator) is bounded by some integer B > 0. This is a family
parameterized by the κ-points (c, d, t) of a smooth Fp -scheme that is determined by B.
For fixed c, d ∈ κ× , the assertions in Theorem 1.1 concerning the fibral root numbers
and the generic rank for the associated elliptic curve in (1.4) are unaffected by replacing κ
with a finite extension. (This is also crucial in the proof of Theorem 1.1; see (4.6) and the
surrounding text.) Thus, granting the parity conjecture, Theorem 1.1 implies that there
is a systematic “rank gap” ≥ 1 between generic and special Mordell–Weil ranks over the
connected components of the family of planar Weierstrass models Et as (c, d, t) and κ vary
with height(t) ≤ B. This is a contrast with a theorem in [14, §9] asserting that there is an
average “rank gap” ≤ 1/2 (or exactly 1/2 under conjectures of Tate) between generic and
4
B. CONRAD, K. CONRAD, AND H. HELFGOTT
special Mordell–Weil ranks of Jacobians of certain universal families of pencils of smooth
plane curves in characteristic p. (The pencils considered in [14, §9] are induced by smooth
surfaces in P1 × P2 , but the closure of (1.4) in P1F ×F P2F is not F -smooth.)
Here is an overview of how we prove Theorem 1.1. To compute the rank of a Mordell–
Weil group, we wish to use a 2-descent, and this is simplest when there is a rational point
of order 2 and we are not in characteristic 2. Weierstrass models for such elliptic curves can
always be brought to the form y 2 = x3 + Ax2 + Bx with the 2-torsion point equal to (0, 0).
Step 1: For an odd prime p, consider the non-isotrivial elliptic curve over Fp (T ) given
by the Weierstrass model with A = T , B = −T 3 :
ET : y 2 = x3 + T x2 − T 3 x.
For every t ∈ κ(u) with t 6= 0, −1/4, the specialization Et may be considered as an elliptic
curve over κ(u). We will compute the reduction type of Et at every place of κ(u). When
char(κ) = 3, the 2-torsion point (0,0) will prevent the intervention of wild ramification.
Step 2: We show that QT = (−T, T 2 ) ∈ ET (Fp (T )) has infinite order, so Qt has infinite
order in Et (κ(u)) for every t ∈ κ(u) such that t 6∈ κ. This uses an extension to characteristic
p of the classical Nagell–Lutz criterion in characteristic 0.
Step 3: Letting h(T ) = cT 2p + du, where c, d ∈ κ× , we use algebraic properties of
the defining Weierstrass model for ET to find a simple formula (3.16) for W (Eh(t) ) for
every t ∈ κ(u). (Note h(t) 6∈ κ for every t.) The formula for W (Eh(t) ) implies that
W (Eh(t2 +u) ) = 1 for all t ∈ κ(u); the elliptic curve Eh(T 2 +u) is (1.4). Our specific choice of
h(T ) is partly motivated by the study of the characteristic-p Möbius function in [4].
Step 4: As just noted, Eη in (1.4) is Eh(T 2 +u) . The point Qh(T 2 +u) on this curve has
infinite order, and we use this point to show that Et (F ) has positive rank for all t ∈
P1 (F ) − {∞}. A mixture of geometric, arithmetic, and cohomological arguments is used
to prove that the rank of Eη (F (T )) is ≤ 1 (so the rank is exactly 1). The essential inputs
are the Lang–Néron theorem over an algebraic closure κ, the geometry of the locus of bad
reduction for Eη over P1 × P1 , and some arithmetic considerations with the Chebotarev
density theorem. Standard geometric upper bounds on the F (T )-rank give very large bounds
when applied to Eη . It therefore seems hopeless to calculate the rank of Eη (F (T )) via purely
geometric methods, even though the generic-rank conclusion in Theorem 1.1 holds over κ.
Steps 1 and 2 are carried out in §2, Step 3 is carried out in §3, and Step 4 is carried out
in §4–§6. The bulk of the work is in Step 4, which the geometrically-inclined reader may
prefer to read directly after Step 1. We note that Steps 2 and 3 are logically independent,
as are Steps 3 and 4. (Clearly Step 2 is used in Step 4.) In §7, we discuss a rank conjecture
of Nagao in the context of (1.4).
In Appendix A, we review previous work on variation of root numbers in families over
Q. The reason to expect the possibility of different behavior in positive characteristic is
explained in Appendix B. These appendices are expository, but they should help the reader
to have the proper perspective on our work.
Our notation is standard, with two exceptions: F denotes the rational function field κ(u),
where κ is a finite field that is assumed to have characteristic 6= 2 unless otherwise stated,
and in some calculations in a field we shall use the shorthand x ∼ y to denote the relation
x = yz 2 for a non-zero z (see Definition 3.2).
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
5
2. Reduction type and rank for y 2 = x3 + T x2 − T 3 x
We begin with two elementary lemmas concerning reduction types for an elliptic curve
over the fraction field of a discrete valuation ring. We write K for the fraction field and
v for the normalized (i.e., Z-valued) discrete valuation on K, with valuation ring OK and
residue field k. Both lemmas are standard when char(k) 6= 2, 3, so a key point of the proofs
is to include the case char(k) = 3. For an elliptic curve E over K, we let ∆, c4 , and c6
denote the usual parameters associated to a Weierstrass model for E over K. (As is wellknown, ∆ mod (K× )12 , c4 mod (K× )4 , and c6 mod (K× )6 are independent of the choice of
Weierstrass model.)
Lemma 2.1. Let E be an elliptic curve over K with potentially good reduction. If there is
good reduction, then v(∆) ≡ 0 mod 12. The converse holds in either of the two following
situations:
(1) char(k) 6= 2, 3,
(2) char(k) 6= 2 and E(K)[2] 6= O.
Proof. The necessity of the congruence condition for good reduction is obvious. When
char(k) 6= 2, 3, all integral Weierstrass models can be put in the form y 2 = x3 + αx + β,
so the sufficiency of the congruence condition for good reduction in case (1) is proved by
direct calculation using the standard formulas for j and ∆ (in terms of α and β) and using
the coordinate changes (x, y) 7→ (γ 2 x, γ 3 y) with γ ∈ K× .
For sufficiency in case (2), we may suppose OK is strictly henselian. Thus, k is separably
closed with char(k) 6= 2, so ∆ must be a square in K× since v(∆) is even. By the 2-torsion
hypothesis, for any Weierstrass K-model y 2 = f (x) for E there is at least one K-rational
root of the cubic f . The discriminant of f is a square in K× , so f splits over K and hence
E[2] is K-split.
Let Ksep /K be a separable closure and let Γ ⊆ GL2 (Z2 ) be the image of the 2-adic
representation
ρE,2 : Gal(Ksep /K) → Aut(lim E[2n ](Ksep )) ' GL2 (Z2 )
←−
attached to E. Since E[2] is K-split, Γ has trivial reduction modulo 2. Thus, the Galois
action must be pro-2, and hence tame. Since E has potentially good reduction and OK
is strictly henselian, Γ must be a finite cyclic 2-group. Pick γ ∈ Γ, so γ = 1 + 2x with
x ∈ M2 (Z2 ). Since the 2-adic cyclotomic character over K is trivial,
1 = det(γ) = 1 + 2Tr(x) + 4 det(x) = −1 + Tr(γ) + 4 det(x).
In particular, Tr(γ) ≡ 2 mod 4. Elements in GL2 (Q2 ) with order 4 have characteristic
polynomial X 2 + 1 and thus have trace 0. This shows that Γ cannot contain elements of
order 4, so Γ is either trivial or has order 2. Hence, if K0 ⊆ Ksep is the splitting field for
ρE,2 then [K0 : K] divides 2.
The quadratic twist E 0 of E by K0 /K must have trivial 2-adic representation, and hence
it has good reduction. Since OK is strictly henselian and [K0 : K] divides 2,
v(∆(E 0 )) ≡ v(∆(E)) + 12/[K0 : K] ≡ 12/[K0 : K] mod 12
and v(∆(E 0 )) ≡ 0 mod 12 (since E 0 has good reduction). Thus, [K0 : K] = 1, so E ' E 0 has
good reduction.
Lemma 2.2. Assume char(k) 6= 2 and let E be an elliptic curve over K with potentially
multiplicative reduction. The parameters c4 and c6 are non-zero, and there is multiplicative
6
B. CONRAD, K. CONRAD, AND H. HELFGOTT
reduction if and only if v(c4 ) ≡ 0 mod 4. When OK is complete, there is split multiplicative
reduction if and only if −c6 is a square in K× .
Proof. By hypothesis, j is non-integral. As is well-known, c4 and c6 are non-zero when j 6=
0, 1728. The formation of the Néron model commutes with base change to the completion, so
we may suppose OK is complete. Since 2 ∈ K× , the quadratic extensions of K are classified
×
by K× /(K× )2 , and since 2 ∈ OK
, the unramified quadratic extensions of K are classified by
×
(a)
the unit classes (modulo unit squares). For
√ a ∈ K , let E denote the quadratic twist of
E by the non-trivial character of Gal(K( a)/K). Clearly
c4 (E (a) ) ≡ a2 c4 (E) mod (K× )4 , c6 (E (a) ) ≡ a3 c6 (E) mod (K× )6 .
By the theory of Tate models, there is a unique class u = u(E) modulo (K× )2 such that
has split multiplicative reduction. Moreover,
• E has multiplicative reduction if and only if u is a unit class in K× /(K× )2 ,
• E has split multiplicative reduction if and only if u is trivial in K× /(K× )2 .
E (u)
A direct calculation with Tate models shows c4 (E (u) ) ∈ (K× )4 and −c6 (E (u) ) ∈ (K× )2
×
(since 2 ∈ OK
). We conclude that
−c6 (E) ≡ u mod (K× )2 ,
so the reduction is split multiplicative if and only if −c6 (E) is a square in K× . Also,
u ∈ K× /(K× )2 is a unit class if and only if v(u) ≡ 0 mod 2, or equivalently v(u2 ) ≡ 0 mod 4.
Therefore, since
v(c4 (E)) ≡ v(c4 (E (u) )) − v(u2 ) ≡ −v(u2 ) mod 4,
the reduction is multiplicative if and only if v(c4 (E)) ≡ 0 mod 4.
Since we are interested in working with elliptic curves that are not in characteristic 2 and
have a non-zero rational 2-torsion point, the shape of a Weierstrass model can be taken to
be
(2.1)
E : y 2 = x3 + Ax2 + Bx.
The discriminant ∆ and parameters c4 and c6 of such a model are given by the following
formulas [34, p. 46]:
(2.2)
∆ = 16B 2 (A2 − 4B), c4 = 16(A2 − 3B), c6 = −32A(2A2 − 9B).
For P = (x, y) ∈ E − E[2], the point [2]P has coordinates given by [34, pp. 58–59]:
!
2
2
2
x −B
3x2 + 2Ax + B x2 − B
x3 − Bx
,−
+
(2.3)
[2]P =
.
2y
2y
2y
2y
We set A = T and B = −T 3 in (2.1), giving the elliptic curve
(2.4)
ET : y 2 = x3 + T x2 − T 3 x
over Fp (T ) with p 6= 2. By (2.2), the discriminant and j-invariant of (2.4) are
(2.5)
∆ = 16T 8 (1 + 4T ),
j=
c34
256(1 + 3T )3
= 2
,
∆
T (1 + 4T )
and the parameters c4 and c6 are
(2.6)
c4 = 16T 2 (1 + 3T ),
c6 = −32T 3 (2 + 9T ).
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
7
v(t)
Reduction Type
> 0, even
multiplicative
> 0, odd
(pot. mult.) additive
< 0, ≡ 0 mod 4
good
< 0, 6≡ 0 mod 4
(pot. good) additive
= 0, v(1 + 4t) = 0
good
= 0, v(1 + 4t) > 0
multiplicative
Table 1. Reduction types for Et , t ∈ F − κ
For each t ∈ F = κ(u) with t 6∈ κ, the Weierstrass model
Et : y 2 = x3 + tx2 − t3 x
(2.7)
defines an elliptic curve over F . (If t ∈ κ − {0, −1/4} then Et is also an elliptic curve over
F , but assuming t 6∈ κ will avoid some unnecessary complications.)
Theorem 2.3. Fix t ∈ F = κ(u) with t 6∈ κ. Let v be a place on F . The reduction type of
Et at v is as in Table 1.
Proof. Specializing (2.5) and (2.6), the parameters of Et are
(2.8)
∆ = 16t8 (1 + 4t), j =
256(1 + 3t)3
, c4 = 16t2 (1 + 3t), c6 = −32t3 (2 + 9t).
t2 (1 + 4t)
(We write ∆ instead of ∆|T =t , and likewise for the other parameters.) None of the parameters in (2.8) is 0, since t 6∈ κ.
If v(t) > 0, then
v(∆) = 8v(t), v(c4 ) = 2v(t), v(j) = −2v(t) < 0,
so there is potentially multiplicative reduction. Using Lemma 2.2, there is multiplicative
reduction when v(t) is even, and there is additive reduction when v(t) is odd.
If v(t) < 0 and char(κ) > 3, then
v(∆) = 9v(t), v(c4 ) = 3v(t), v(j) = 0,
so there is potentially good reduction. If v(t) < 0 and char(κ) = 3, then
v(∆) = 9v(t), v(c4 ) = 2v(t), v(j) = −3v(t) > 0,
so again there is potentially good reduction. Using Lemma 2.1 in both cases, there is good
reduction when v(t) ≡ 0 mod 4 and there is additive reduction otherwise.
Finally, suppose v(t) = 0, so
v(∆) = v(1 + 4t),
v(c4 ) = v(1 + 3t).
Both 1 + 4t and 1 + 3t have non-negative valuation at v, and the valuations are not both
positive. If v(1 + 4t) = 0 then v(j) = 3v(c4 ) ≥ 0, so there is good reduction (by Lemma
2.1). If v(1 + 4t) > 0 then v(c4 ) = 0, so v(j) = −v(∆) < 0. This implies (by Lemma 2.2)
that there is multiplicative reduction at v in such cases.
Now we turn to the Mordell–Weil group of the generic fiber, ET (Fp (T )). As before,
p 6= 2. Two obvious non-zero rational points are (0, 0) and Q = (−T, T 2 ). (There is another
obvious non-zero rational point, (T 2 , T 3 ), but this is (0, 0) + Q.) We will prove that Q has
infinite order, so rank(ET (Fp (T ))) ≥ 1.
8
B. CONRAD, K. CONRAD, AND H. HELFGOTT
For elliptic curves over Q, explicit rational points are usually checked to be non-torsion
by the Nagell–Lutz integrality criterion. This criterion is really a collection of local criteria
over Z(p) for all primes p. We need an analogue for discrete valuation rings with positive
characteristic. Here is a version over arbitrary discrete valuation rings.
Theorem 2.4. Let R be a discrete valuation ring with residue field k of characteristic
p ≥ 0, and let K be its fraction field. Let E/K be an elliptic curve, and let P ∈ E(K) be a
non-zero torsion point.
If there exists a Weierstrass model of E over R such that one of the affine coordinates of
P does not lie in R, then the scheme-theoretic closure of hP i ⊆ E(K) in the Néron model
of E over R is a finite flat local R-group. In particular, p > 0 and P has p-power order. If
in addition char(K) = p, then E/K has potentially supersingular reduction and j(E) ∈ K is
a pth power.
It follows from the Oort–Tate classification and Cartier duality that the only example of
a non-trivial finite flat local group scheme over Zsh
(p) with p-power order and cyclic constant
generic fiber is µ2 for p = 2. Thus, Theorem 2.4 recovers the integrality of non-trivial
torsion points on Weierstrass Z-models of the form y 2 = f (x) (for which non-zero 2-torsion
points must have the form (x0 , 0) with x0 ∈ Z).
Proof. Let W ⊆ P2R be the chosen Weierstrass R-model for E (so there is a chosen isomorphism WK ' E as pointed curves over K). Let W sm ⊆ W be the open R-smooth locus of
W , and let ε ∈ W (R) = W (K) = E(K) be the section [0, 1, 0], so ε ∈ W sm (R). Since P
viewed as a point
Pe ∈ W (K) − {[0, 1, 0]} ⊆ A2 (K) = K × K
is assumed to have at least one coordinate not in R, as a point of P2 (K) = P2 (R) its
reduction in P2 (k) cannot lie in A2k . Therefore, the reduction must lie on the line at
infinity. However, by the theory of Weierstrass models we know that Wk has εk as its
unique point on this line, so the reduction of Pe is εk . Since εk ∈ Wksm , we conclude that
Pe ∈ W sm (R).
Let E be the Néron model of E over R. By the Néron mapping property, there is a
unique map W sm → E over R extending the identification of K-fibers with E. This map
carries ε to the identity element in E (R), so the image of Pe in E (R) reduces to the identity
in Ek . In other words, under the equality E(K) = E (R), the reduction of P in the closed
fiber Ek of the Néron model must be the identity.
Let N > 1 be the order of P . By the Néron mapping property, P defines a map of
R-groups Z/N Z → E that is a closed immersion on the generic fiber. Since the R-group
Z/N Z is proper and the target E is separated over the Dedekind domain R, the schemetheoretic image of this map is a finite flat R-subgroup G ,→ E with order N and constant
generic fiber; this must be the closure of hP i. The closed fiber of G must be infinitesimal
since P has reduction equal to the identity. This forces G to be local, so the characteristic
p of k must be positive and the order N of G must be a power of p.
Now assume char(K) = p. We must prove that j(E) is a pth power in K and that E has
potentially supersingular reduction. To prove that j(E) is a pth power in K when E(K)
contains a non-trivial point with p-power order, we use the classical fact that if L is a field
with characteristic p > 0 and E is an elliptic curve over L such that there exists an étale
subgroup Γ ⊆ E with order pn for some n ≥ 1 (that is, E is ordinary and the connectedétale sequence of E[pn ] splits over L), then j(E) ∈ L is a pn th power in L. To prove this
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
9
fact, let E 0 = E/Γ, so the isogeny E 0 → E that is dual to the projection E → E 0 has kernel
that is Cartier-dual to Γ. This kernel is therefore multiplicative with p-power order, so it
is infinitesimal. Since E 0 is a 1-dimensional abelian variety in characteristic p, it contains
n
a unique infinitesimal subgroup of order pn . The relative n-fold Frobenius E 0 → E 0 (p ) has
n
this subgroup as its kernel, so the two quotients E and E 0 (p ) of E 0 are L-isomorphic as
n
n
quotients of E 0 . In particular j(E) = j(E 0 (p ) ) = j(E 0 )p in L.
Finally, returning to our initial situation, we must show that E has potentially supersingular reduction if K has characteristic p > 0. The assumptions on R and on the coordinates
of P are unaffected by replacing K with a finite separable extension K0 and replacing R
with a maximal-adic localization of its integral closure in K0 , so we may assume that E
has semistable reduction. It must be proved that the Néron model E in this case has fibral
identity component Ek0 that is a supersingular elliptic curve. Assume to the contrary, so Ek0
is either a torus or an ordinary elliptic curve; we seek a contradiction. In either case, the
finite local subgroups of Ek0 are multiplicative. Hence, if we construct G as we did above
(the scheme-theoretic closure of hP i in E ) then the infinitesimal closed fiber Gk ,→ Ek must
lie in Ek0 , so Gk is multiplicative with p-power order. Since the generic fiber GK is constant,
we conclude that the Cartier dual G∨ has multiplicative generic fiber. However, G∨ is finite
∨
∨
and flat over R with special fiber G∨
k that is étale, so G is R-étale. This forces GK to be
both multiplicative and étale, but a non-zero étale K-group with p-power order cannot be
multiplicative when K has characteristic p, so we have reached a contradiction.
Corollary 2.5. With notation as in Theorem 2.4, if char(K) = p > 0 and j(E) ∈ K is not
a pth power, then non-zero torsion points in E(K) have integral coordinates with respect to
any Weierstrass R-model of E.
As an application of Corollary 2.5, we have:
Corollary 2.6. Assume p 6= 2. The Fp (T )-rational point Q = (−T, T 2 ) on the elliptic
curve ET in (2.4) has infinite order.
In particular, for any field L of characteristic p and any t ∈ L that is transcendental over
Fp , the specialization Qt ∈ Et (L) that is obtained by sending Fp (T ) into L by T 7→ t is a
point of infinite order.
Proof. The second claim follows from the first since the field extension Fp (T ) → L defined
by T 7→ t induces an injection of groups ET (Fp (T )) → Et (L).
To see that Q has infinite order in ET (Fp (T )), first note the j-invariant of ET , as given
in (2.5), is not a pth power in Fp (T ). Therefore, by Corollary 2.5, an Fp (T )-rational point
on ET has infinite order provided that, using the Weierstrass model (2.4) for ET , some
non-zero multiple of the point has an x- or y-coordinate that is non-integral at a finite place
on Fp (T ). (The Weierstrass model (2.4) is integral away from ∞.)
Since x(Q) and y(Q) are integral away from ∞ and y(Q) 6= 0, we double Q. By (2.3),
!
(T + 1)(T 2 − 4T − 1)
T +1 2
,
.
[2](Q) =
2
8
Thus, x([2]Q) and y([2]Q) are integral away from ∞ and y([2]Q) 6= 0, so we double again
and find
2
(T + 1)4 + 16T 3
x([4](Q)) =
.
4(T + 1)(T 2 − 4T − 1)
This is non-integral at the place T + 1, so we are done.
10
B. CONRAD, K. CONRAD, AND H. HELFGOTT
Remark 2.7. By the Lang–Néron theorem, the group ET (Fp (T )) is finitely generated.
This group has rank at least 1, since we have exhibited an explicit element with infinite
order. Theorem 1.1 implies that ET (L) has rank 1 for certain extensions L of Fp (T ) with
transcendence degree 2 over Fp , so a posteriori we conclude that ET (Fp (T )) has rank 1.
Presumably the proof of Theorem 1.1 can be modified to give a direct proof that ET (Fp (T ))
has rank 1, without requiring the use of such auxiliary fields L.
3. Root numbers
For ET as in (2.4), we will compute the local root numbers Wv (Et ) for t ∈ κ(u) with
t 6∈ κ. Let us first collect a general list of local root number formulas. This is well-known
for residue characteristic p 6= 2, 3, but we include some cases with p = 3.
Theorem 3.1. Let K be a local field, with finite residue field of characteristic p 6= 2 and
normalized valuation v : K× → Z. Let χK be the quadratic character of the residue field of
K, and let E be an elliptic curve over K.
(1) Assume E has potentially good reduction, and if p = 3 then assume E(K)[2] 6= O.
Define e = 12/ gcd(v(∆), 12). We have e ∈ {1, 2, 3, 4, 6}, with 3 - e when p = 3, and
the local root number WK (E) can be computed by the following formulas:

if e = 1,

1

χ (−1) if e = 2 or 6,
K
WK (E) =

χ
if e = 3,
K (−3)



χK (−2) if e = 4.
(2) Suppose E has potentially multiplicative reduction. If the reduction is additive then
WK (E) = χK (−1). If the reduction is multiplicative and c6 = c6 (E) is computed
using any Weierstrass K-model of E, then WK (E) = −1 when −c6 is a square in
K× and WK (E) = 1 when −c6 is not a square in K× .
Proof. We first address the properties of e in the cases with potentially good reduction. By
the method of proof of Lemma 2.1(2), if E has potentially good reduction then it acquires
good reduction over a quadratic extension of a splitting field for E[2]. This splitting field is
a tame Galois extension with degree dividing 6, so Lemma 2.1 implies that e must divide
12 and moreover that if p = 3 then 3 - e. The tameness and Lemma 2.1 ensure that e
is the order of the image of inertia in the `-adic representation for E (any ` 6= p). The
cyclicity of tame inertia therefore rules out the possibility e = 12, since there are infinitely
many rational primes ` > 3 for which the 12th cyclotomic polynomial Φ12 has no quadratic
factors over Q` .
Before we treat the general case, let us consider the special case K = Qp with p 6= 2. In
this case, the proposed formulas are proved by Rohrlich for p > 3 in [28, Prop. 2] when the
reduction is potentially good and (using Lemma 2.2) in [28, Prop. 3] when the reduction is
potentially multiplicative. (Also see [29, Prop. 3] for further discussion in the multiplicative
case.) By Lemma 2.1(2) and Lemma 2.2, the proofs of [28, Prop. 2, 3] work in our cases
when p = 3. (The additional 2-torsion hypothesis in potentially good reduction cases for
p = 3 avoids wild ramification.)
Rohrlich’s proofs in [28] and [29] are representation-theoretic and rest on papers of
Deligne [6] and Tate [36] that are valid for local fields with any residual (or generic) characteristic. Thus, these proofs carry over to the general case (with residue characteristic
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
11
v(t)
Wv (Et )
> 0, even
hard to use
> 0, odd
χv (−1)
< 0, even
χv (−1)v(t)/2
< 0, odd
χv (−2)
= 0, v(1 + 4t) = 0
1
= 0, v(1 + 4t) > 0
−χv (2)
Table 2. Local root numbers on Et , t ∈ F − κ
6= 2, and with a non-trivial rational 2-torsion point in potentially good cases when p = 3).
The “p” in most of the arguments in [28] and [29] should be replaced with the size of the
residue field of K, say q, and the Legendre symbol ( ap ) should be replaced with the Kronecker symbol ( aq ). (Note that when q is an odd prime prime power and a ∈ Z is prime to q,
( aq ) = χK (a).) A general discussion in the context of local and global fields of characteristic
0 may also be found in [30].
Using Theorems 2.3 and 3.1, we now compute the local root numbers Wv (Et ) for every
t ∈ F = κ(u) with t 6∈ κ. Let χ be the quadratic character of κ and let χv be the quadratic
character of the residue field at v. For a ∈ κ, we have χv (a) = χ(a)deg v , where deg v is the
degree of the residue field of v over κ. (Thus, χv = χ when deg v = 1.) Table 2 summarizes
the results, and we will see why the first row is undesirable.
The second, third, fourth, and fifth rows are cases of additive or good reduction (by Table
1 in Theorem 2.3), and these are left to the reader to check via Theorem 3.1. (The third
row is the union of two cases from Table 1 with v(t) < 0, namely v(t) ≡ 0, 2 mod 4. These
two cases are checked separately.) It remains to compute Wv (Et ) in two cases: (i) v(t) is
positive and even, and (ii) v(1 + 4t) > 0. Both are cases of multiplicative reduction, so
Theorem 3.1 requires us to check if −c6 (Et ) is a square in the multiplicative group Fv× of
the completion of F at v. Let us first introduce some convenient notation.
Definition 3.2. Let L be a field. For x, y ∈ L, write x ∼ y when x = yz 2 for some z ∈ L× .
When v(t) is positive and even, in Fv× we compute from (2.6) at T = t that
−c6 = 32t3 (2 + 9t)
∼ 2t(2 + 9t)
∼ t since v(t) > 0.
Thus, by Theorem 3.1(2), Wv (Et ) = −1 if t is a square in Fv× and Wv (Et ) = 1 otherwise.
The last case is v(1 + 4t) > 0. In Fv× ,
−c6 = 32t3 (2 + 9t)
∼ 2t(2 + 9t)
∼ 2t2
∼ 2.
since v(1 + 4t) > 0
Thus, by Theorem 3.1(2), the last entry in Table 2 is confirmed:
(3.1)
v(1 + 4t) > 0 =⇒ Wv (Et ) = −χv (2).
12
B. CONRAD, K. CONRAD, AND H. HELFGOTT
v(t)
Wv (Et2 )
>0
−1
<0
χv (−1)v(t)
2
= 0, v(1 + 4t ) = 0
1
= 0, v(1 + 4t2 ) > 0
−χv (2)
Table 3. Local root numbers on Et2 , t ∈ F − κ
The Et ’s do not have easily manageable global root numbers. There are two main problems. First, the local root number in the first row of Table 2 depends on whether or not t
is a square in Fv× , and that is not something we can easily control. Second, the last row
in Table 2 introduces systematic minus signs. To appreciate the nature of these difficulties,
and how we can avoid them by a change of variables that is peculiar to characteristic p,
let us first try to eliminate the difficulties in the first row of Table 2 by forcing “t” to be a
square: we study the elliptic curve ET 2 . Table 2 is easily translated into this context, and
the results are collected in Table 3. The systematic minus signs in the first and last rows
of Table 3 will cause serious problems.
Write t = g1 /g2 , where g1 , g2 ∈ κ[u] are non-zero and relatively prime. The product of
the local root numbers Wv (Et2 ) over all v yields the global root number formula
Y
Y
Y
(−1) ·
W (Et2 ) = W∞ (Et2 )
χv (−1)v(t) ·
(−χv (2))
v6=∞
v(t)>0
v6=∞
v(t)<0
v6=∞
v(1+4t2 )>0
P
= W∞ (Et2 ) · (−1)#{π:π|g1 } · χ(−1)
π|g2 (deg π) ordπ (g2 )
2
2
·
Y
(−χ(2)deg π )
π|(4g12 +g22 )
P
= W∞ (Et2 ) · (−1)#{π:π|g1 }+#{π:π|(4g1 +g2 )} · χ(−1)deg g2 · χ(2)
2 +g 2 )
π|(4g1
2
deg π
,
where π runs over monic irreducibles in κ[u]. This formula is unwieldy because we cannot
2
2
control the parity of the number
P of irreducible factors of g1 or 4g1 + g2 as t varies. We2 also2
cannot control the parity of π|(4g2 +g2 ) deg π because it is hard to determine when 4g1 + g2
1
2
is separable, though this second problem could be eliminated if we only consider κ in which
χ(2) = 1. Studying Et2 is not helping us to get constant global root numbers at most t (as
is essentially necessary in any example of elevated rank).
Instead of merely simplifying the first row of Table 2 by replacing t with t2 in Et , we
need to eliminate the intervention of the first row of Table 2. To accomplish this, we will
introduce a change of variables in t such that the numerator is always squarefree, and thus
in particular never has positive even valuation at places of F . We also need to acquire
control over the product of minus signs contributed from the last row of Table 2, and this
will be achieved by arguments that are peculiar to positive characteristic.
A “squarefree” change of variables is impossible in characteristic 0, but the pth power
map provides a mechanism to find such a change of variables in characteristic p. The basic
idea is that, for all t ∈ F = κ(u), tp + u has a squarefree numerator and has a pole at ∞,
and thus, for all places v of F , v(tp + u) is never both positive and even. With this noted,
define
(3.2)
h(T ) = cT 2p + du,
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
13
where c, d ∈ κ× . The use of the exponent 2p instead of p will create a counterexample
to Chowla’s two-variable conjecture over κ[u] (see the appendices for a discussion of this
conjecture and its relevance to the study of elevated rank); in concrete terms, this even
exponent will force certain otherwise unknown non-zero quantities we meet later to be
squares. The role of c and d in h(T ) is to provide us with the family of examples in
Theorem 1.1 for each p 6= 2, rather than just one example for each p 6= 2. (The reader may
take c = d = 1 throughout.)
For h(T ) as in (3.2), consider the elliptic curve Eh(T ) over F (T ), obtained by replacing
T with h(T ) in (2.4). We can run through all of our previous work with h(t) in place of t
(rather than t2 in place of t), and now t can be any element of F since h(t) 6∈ κ for all t ∈ F .
(Table 3 only lists root numbers in fibers over t ∈ F − κ.) We will compute W (Eh(t) ) for
all t ∈ F .
Write t = g1 /g2 , where g1 , g2 ∈ κ[u] are relatively prime with g2 6= 0, so
(3.3)
h(t) =
cg12p + dug22p
g22p
.
Call the numerator and denominator, respectively, f1 and f2 :
f1 = cg12p + dug22p ,
(3.4)
f2 = g22p .
Obviously f1 , f2 6= 0. Since (g1 , g2 ) = 1, clearly (f1 , f2 ) = 1. Moreover, since f10 = dg22p =
df2 , f1 is squarefree. Thus, for all finite places v of F , v(h(t)) is never both positive and
even at v. Since
(
−1
if ord∞ (t) ≥ 0,
2p
(3.5)
ord∞ (h(t)) = ord∞ (ct + du) =
2p ord∞ (t) if ord∞ (t) < 0,
we see h(t) has a pole at ∞ for every t ∈ F .
We begin computing local root numbers for Eh(t) over F by starting with the place at
∞, where χ∞ = χ. Using (3.5) and Table 2 (with h(t) in place of t),
(
χ(−2),
if ord∞ (t) ≥ 0,
(3.6)
W∞ (Eh(t) ) =
ord
(t)
∞
χ(−1)
, if ord∞ (t) < 0.
Now let v be a finite place on F . Since h(t) has a squarefree numerator, we get from
Table 2 that
v 6= ∞, v(h(t)) > 0 =⇒ v(h(t)) = 1 =⇒ Wv (Eh(t) ) = χv (−1) = χ(−1)deg v .
Since h(t) has a perfect square g22p as its denominator, Table 2 implies
v 6= ∞, v(h(t)) < 0 =⇒ Wv (Eh(t) ) = χv (−1)v(h(t))/2 = χ(−1)(deg v)·v(g2 ) .
If v(h(t)) = 0, then Table 2 (with h(t) in place of t) tells us that if v(1 + 4h(t)) = 0 then
Wv (Eh(t) ) = 1, whereas
v(1 + 4h(t)) > 0 =⇒ Wv (Eh(t) ) = −χv (2) = −χ(2)deg v .
Combining all of this local information, for t ∈ F the global root number W (Eh(t) ) is
Y
Y
Y
(3.7)
W∞ (Eh(t) )
χ(−1)deg v
χ(−1)(deg v)·v(g2 )
(−χ(2)deg v ).
v6=∞
v(h(t))>0
v6=∞
v(h(t))<0
v6=∞
v(1+4h(t))>0
14
B. CONRAD, K. CONRAD, AND H. HELFGOTT
Referring back to Table 1 with h(t) in place of t, the local root numbers for Eh(t) at places
of multiplicative reduction appear in (3.7) as the terms in the last product.
Writing (3.7) in terms of the numerator and denominator of h(t),
Y
Y
Y
(−χ(2)deg π )
χ(−1)(deg π) ordπ (g2 )
χ(−1)deg π
W (Eh(t) ) = W∞ (Eh(t) )
= W∞ (Eh(t) )
Y
deg π
χ(−1)
= W∞ (Eh(t) )
Y
(deg π) ordπ (g2 )
χ(−1)
deg π
χ(−1)
Y
(−χ(2)deg π )
π|(4f1 +f2 )
π|g2
π|f1
Y
π|(4f1 +f2 )
π|f2
π|f1
deg g2
· χ(−1)
Y
·
(−χ(2)deg π ).
π|(4f1 +f2 )
π|f1
Set
(3.8)
f = 4f1 + f2 = 4cg12p + (4du + 1)g22p .
Since (f, f 0 ) = 1, f is squarefree. We already saw that f1 is squarefree as well, so our global
root number formula simplifies to
(3.9)
W (Eh(t) ) = W∞ (Eh(t) )χ(−1)deg f1 χ(−1)deg g2 µ(f )χ(2)deg f ,
where µ is the Möbius function on κ[u] (defined much like its classical counterpart over Z).
Remark 3.3. Let us clarify how this calculation is analogous to what is seen in work over
Q(T ). Let the Liouville function λ on κ[u] be the totally multiplicative function whose
value on irreducible elements is −1, so if f is separable (i.e., is squarefree) in κ[u] then
µ(f ) = λ(f ). In (3.9), we therefore have an appearance of λ(f ) as a contribution from
local root numbers at places of multiplicative reduction. As is explained in Appendix A,
the Liouville function on Z arises in a similar manner in the study of average root numbers
for elliptic curves over Q(T ) that have a point of multiplicative reduction on P1Q . Another
similarity with the situation in characteristic 0 is that λ is being computed on an element
f ∈ κ[u] that is the value at (g1 , g2 ) of a homogeneous 2-variable polynomial over κ[u],
where g1 and g2 are relatively prime. (Consider g1 and g2 in (3.8) as specializations of
independent indeterminates over κ[u].) Compare this with the appearance of λ(fE (m, n))
in the discussion at the end of Appendix A.
To simplify (3.9) further, we compute the degrees in the exponents. This depends on the
relative sizes of deg g1 and deg g2 . Let
n1 = deg g1 ,
n2 = deg g2 ,
with the standard convention n1 = −∞ when g1 = 0, so
(
2pn2 + 1 if n1 ≤ n2 ,
(3.10)
deg f1 =
deg f2 = 2pn2 .
2pn1
if n1 > n2 ,
By inspection, deg f1 > deg f2 , so
(3.11)
(
2pn2 + 1
deg f =
2pn1
if n1 ≤ n2 ,
if n1 > n2 .
Using (3.6), (3.9), (3.10), and (3.11),
(
χ(−2)χ(−1)χ(−1)n2 µ(f )χ(2)
W (Eh(t) ) =
χ(−1)n2 −n1 · 1 · χ(−1)n2 · µ(f ) · 1
if n1 ≤ n2 ,
if n1 > n2 ,
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
15
so
(3.12)
(
χ(−1)n2 µ(f )
W (Eh(t) ) =
χ(−1)n1 µ(f )
if n1 ≤ n2 ,
if n1 > n2 ,
where t = g1 /g2 ∈ κ(u) is expressed in reduced form and f is defined in (3.8). (The two
cases in (3.12) are classified by the sign of ord∞ (t) = n2 − n1 .)
To complete the calculation of W (Eh(t) ) for t ∈ F , we need to compute µ(f ). For this,
we use a remarkable fact: the Möbius function in characteristic p is a more accessible object
than its classical counterpart over Z. Indeed, there is a formula for the Möbius function on
κ[u] other than its definition. In particular, the explicit calculation of
(3.13)
µ(f ) = µ(4cg12p + (4du + 1)g22p ),
where g1 and g2 appear through their pth powers, can be done without factoring. (Nothing
of the sort can be said for classical variants such as µZ (m2 + 5n2 ).)
The alternative Möbius formula (in Lemma 3.4 below) uses discriminants, so to avoid
any possible confusion on signs and scalar factors, let us briefly recall how to define the
discriminant of a polynomial. For any field K and any non-zero polynomial P = P (u) in
K[u] with degree n, the discriminant of P is
(3.14)
n(n−1)/2
discK P := (−1)
n−2
· (lead P )
·
n
Y
P 0 (γi ) ∈ K,
i=1
where γ1 , . . . , γn are the roots of P (repeated with multiplicity) in a splitting field and
lead P ∈ K × is the leading coefficient of P . Obviously discK (cP ) = c2n−2 · discK (P )
for c ∈ K × . (In [4], which motivated the work in this section, a different definition of
the discriminant is used that is invariant under K × -scaling of P . That definition differs
from (3.14) by an even power of lead P . Discriminants will only matter up to a non-zero
square scaling factor for our purposes, because of the quadratic character in (3.15) below,
so the different discriminants used in [4] and here are not incompatible for the intended
applications.)
Lemma 3.4. Let κ be a finite field with odd characteristic, and let χ be the quadratic
character on κ, with χ(0) = 0. For any non-zero polynomial P ∈ κ[u],
(3.15)
µ(P ) = (−1)deg P χ(discκ P ),
where discκ P is the discriminant of the polynomial P .
Proof. This formula is trivial when P has a multiple factor: both sides are 0. When P is
separable (that is, squarefree) and has r prime factors, (3.15) is the same as: χ(discκ P ) =
(−1)deg P −r . Written this way, (3.15) appears in [35, Cor. 1]. The properties of finite fields
that are most essential in the proof of [35, Cor. 1] are perfectness and pro-cyclicity of their
Galois theory. (The only reason to assume char(κ) 6= 2 is that the Möbius formula can then
be given in terms of the quadratic character; a formula when char(κ) = 2 can be found in
[4] and [35], but it uses a lift to characteristic 0. We omit this formula since we do not need
it.)
Direct computation of polynomial discriminants can often be unwieldy, so applications
of Lemma 3.4 usually rest on the connection between discriminants and resultants (see
[4] and [35]); such work with resultants requires special care in positive characteristic. In
16
B. CONRAD, K. CONRAD, AND H. HELFGOTT
our specific situation we will be able to extract the required information directly from the
definition of the discriminant, so we will not need to use resultants.
In (3.12) we have to compute µ(f ) for f = 4cg12p + (4du + 1)g22p such that g2 6= 0 and
(g1 , g2 ) = 1. The peculiar coefficient of g22p is an artifact of our elliptic curve Eh(T ) . We
shall carry out the Möbius calculation for a cleaner expression and then return to µ(f ).
Lemma 3.5. Let κ be a finite field with characteristic p 6= 2. Using the convention deg(0) =
−∞, for a, b ∈ κ× and relatively prime g1 , g2 ∈ κ[u] we have
(
−χ(−1)deg g2 if deg g1 ≤ deg g2 ,
2p
2p
µ(ag1 + bug2 ) =
χ(−1)deg g1
if deg g1 > deg g2 .
Proof. The cases when g1 = 0 or g2 = 0 are trivial, so we now
Set g = ag12p + bug22p , n1 = deg g1 , n2 = deg g2 . Since g 0 =
squarefree. We have (with notation as in Definition 3.2)
(
(
2pn2 + 1 if n1 ≤ n2 ,
b
deg g =
lead g ∼
2pn1
if n1 > n2 ,
a
suppose both are non-zero.
bg22p and (g1 , g2 ) = 1, g is
if n1 ≤ n2 ,
if n1 > n2 .
Let n = deg g, so in a splitting field the set of
Q distinct roots of g may be labelled as
{γ1 , . . . , γn }. Since g 0 = bg22p , it follows that i g 0 (γi ) is in κ× and may be computed
modulo squares:
Y
Y
g 0 (γi ) = bdeg g ·
g2 (γi )2p ∼ bdeg g
i
because
Q
i
×
i g2 (γi ) ∈ κ . Hence,
discκ (g) = (−1)n(n−1)/2 (lead g)n−2 ·
n
Y
g 0 (γi ) ∼ (−1)n(n−1)/2 (b · lead g)n .
i=1
Since b · lead g ∼ b2 when n is odd (that is, when n1 ≤ n2 ), we conclude
discκ (g) ∼ (−1)n(n−1)/2 ∼ (−1)max(n1 ,n2 )
by the formula for n. Hence, by (3.15), µ(g) = (−1)n χ(discκ (g)) = (−1)n χ(−1)max(n1 ,n2 ) .
It is now a simple matter to finish the computation of the global root number:
Theorem 3.6. Let h(T ) = cT 2p + du, where c, d ∈ κ× . Let ET be defined as in (2.4). For
any t ∈ F = κ(u), the elliptic curve Eh(t) over F satisfies
(
−1 if ord∞ (t) ≥ 0,
(3.16)
W (Eh(t) ) =
1
if ord∞ (t) < 0.
Proof. Write t = g1 /g2 where g2 6= 0 and (g1 , g2 ) = 1. We may apply Lemma 3.5 to the
polynomial f = 4cg12p + (4du + 1)g22p by making the linear change of variables u 7→ u − 1/4d
that preserves degrees. This yields
(
−χ(−1)n2 , if n1 ≤ n2 ,
(3.17)
µ(f ) =
χ(−1)n1 ,
if n1 > n2 ,
where n1 = deg g1 and n2 = deg g2 (and n1 = −∞ if g1 = 0). Combining (3.17) with the
global root number formula (3.12) yields (3.16).
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
17
To force the global root number to be 1, we want only the second case of (3.16) to occur.
This can be achieved by a simple trick (related to (3.2), but initially inspired by [16, p. 57]):
replace t with t2 + u, which has a pole at ∞ for every t in F = κ(u). Thus,
(3.18)
W (Eh(t2 +u) ) = 1
for every t ∈ F = P1F (F ) − {∞}. Since (1.4) is the Weierstrass model in the definition
of Eh(T 2 +u) , we see that (1.4) is not as arbitrary as it may have initially appeared to be.
Combining (3.18) with Remark 1.2 settles the root number aspect of Theorem 1.1.
4. Generic rank bound I. Specialization at points of height 0
Write (1.4) in the form
(4.1)
Eη : y 2 = x3 + h(T 2 + u)x2 − (h(T 2 + u))3 x,
where h(T ) = cT 2p +du and c, d ∈ κ× . We have shown in §3 that for each t ∈ P1 (F ), Et is an
elliptic curve over F with global root number 1. The elliptic curve Eη over F (T ) = κ(u, T ) is
obtained from ET /Fp (T ) in (2.4) by replacing T with the element h(T 2 +u) ∈ F (T ) = κ(u, T )
that is not in κ, so the point (−T, T 2 ) ∈ ET (Fp (T )) goes over to the point
(4.2)
Q = (−h(T 2 + u), (h(T 2 + u))2 ) ∈ Eη (F (T ))
that has infinite order (Corollary 2.6). For every t ∈ F the specialization h(t2 + u) ∈ κ(u) is
not in κ, so the specialization of Q in Et (F ) must likewise have infinite order for all t ∈ F .
Thus, all specializations Et (F ) at t ∈ P1 (F ) − {∞} have positive rank. This settles the
rank aspect of Theorem 1.1 for the F -rational fibers. (We already noted in Remark 1.2 that
E∞ (F ) has rank 0.)
The remainder of this paper is devoted to proving that the generic Mordell–Weil group
Eη (F (T )), which we know has rank at least 1, has rank exactly 1. This will complete the
proof of Theorem 1.1.
Since the cubic polynomial in x given by the Weierstrass model (4.1) defining Eη is the
product of x and an irreducible quadratic polynomial in F (T )[x], the only nontrivial rational
2-torsion is the point (0, 0). Therefore
(4.3)
dimF2 Eη (F (T ))/2 · Eη (F (T )) = 1 + rank(Eη (F (T ))).
A point of infinite order in Eη (F (T )) is given in (4.2), so the generic rank is 1 if and only
if the dimension in (4.3) is at most 2.
Viewing F (T ) = κ(u, T ) as the function field of P1 × P1 , we shall now consider specialization along the u-line. We will specialize at generic points of {u0 } × P1κ for closed points
u0 ∈ P1κ ; these generic points are identified with the closed points of height 0 on the u-line
P1κ(T ) over κ(T ). For such u0 , let its residue field be written as κ0 = κ(u0 ); this is a finite
field and the notation κ0 will be used constantly in what follows. If u0 6= ∞ then we also
write u0 to denote the image of the indeterminate u under the quotient map κ[u] κ0 .
Using (2.5) and (2.6), the parameters ∆ and c4 for (4.1) are given by
(4.4)
∆ = 16(h(T 2 + u))8 (1 + 4h(T 2 + u)),
c4 = 16(h(T 2 + u))2 (1 + 3h(T 2 + u)).
From the formula for ∆, we see that for all closed points u0 ∈ A1κ , the u0 -specialization
of (4.1) is an elliptic curve over κ0 (T ). The elliptic curves Et for t ∈ P1 (F ) all live over
the fixed global field F = κ(u), but the u0 -specializations Eu0 of Eη live over the global
fields κ0 (T ) = κ(u0 )(T ) that vary. The notation Eu0 presents no risk of confusion with
18
B. CONRAD, K. CONRAD, AND H. HELFGOTT
the notation Et for specialization at t ∈ P1 (F ) because we will never again use such T specializations.
Let us briefly describe a natural but ultimately unsuccessful strategy for using the Eu0 ’s
to show that the dimension in (4.3) is at most 2. We can prove a “height 0” version of
Silverman’s specialization theorem for abelian varieties, and from this it follows that for all
but finitely many height-0 points u0 ∈ P1κ(T ) , the specialization map
Eη (F (T )) → Eu0 (κ0 (T ))
(4.5)
at u0 is injective. Thus, it would suffice to prove rank(Eu0 (κ0 (T ))) ≤ 1 for infinitely many
u0 . For an infinite set of points u0 (specifically, the ones arising from Theorem 5.1 below),
we can prove rank(Eu0 (κ0 (T ))) ≤ 3. (Switching root number calculations to the u0 -side, we
also can show W (Eu0 ) = −1. This suggests, but does not prove, that Eu0 (κ0 (T )) has rank
1 or 3.) For such u0 , the subspace Vu0 of everywhere-unramified classes in the 2-Selmer
group S [2] (Eu0 /κ0 (T ) ) is 2-dimensional, and we can show that rank(Eu0 (κ0 (T ))) = 1 (resp.
< 3) if and only if the natural map Vu0 → X(Eu0 )[2] is injective (resp. non-zero). The
Cassels–Tate pairing of the image of a basis of Vu0 in X(Eu0 )[2] can be calculated by using
a method of Cassels [2], but unfortunately it always turns out to be trivial! Thus, we do
not know how to prove that Vu0 has non-zero image in X(Eu0 )[2] for infinitely many of the
points u0 as in Theorem 5.1, and hence we do not know if rank(Eu0 (κ0 (T ))) < 3 (let alone
if Eu0 (κ0 (T )) has rank 1) for infinitely many u0 .
Here is the successful strategy for using arithmetic information from the Eu0 ’s to bound
the dimension in (4.3). We are aiming to prove that Eη (κ(u, T )) has rank 1, and in (4.2) we
have already found a point of infinite order, so it suffices to bound the rank from above by
1 after replacing κ with a finite extension κ0 that may depend on the parameters c, d ∈ κ×
that were used in the definition of Eη . Since Eη (κ(u, T )) is finitely generated (Lang–Néron),
we may replace κ with a suitable finite extension (depending on c and d) to reduce to the
case when Eη (κ(u, T )) = Eη (κ(u, T )). Now consider the commutative diagram of natural
maps
(4.6)
Eη (κ(u, T ))/2 · Eη (κ(u, T ))
/ Eη (κ(u, T ))/2 · Eη (κ(u, T ))
/ Eu (κ(T ))/2 · Eu (κ(T ))
0
0
Eu0 (κ0 (T ))/2 · Eu0 (κ0 (T ))
in which u0 ∈ A1κ (κ) is a choice of geometric point over a closed point u0 ∈ A1κ , and both
vertical maps are defined by the valuative criterion for properness. Since we adjusted κ so
that Eη (κ(u, T )) = Eη (κ(u, T )), the top side of (4.6) is an isomorphism. Therefore (4.3) is
at most 2 if
• the right side of (4.6) is injective for all but finitely many κ-points u0 ∈ A1κ (κ),
• the image of the map along the bottom side of (4.6) is at most 2-dimensional for
infinitely many closed points u0 ∈ A1κ (equipped with one of the finitely many
choices of κ-point u0 over u0 ).
We consider these two respective assertions as “geometric” and “arithmetic” in nature.
Remark 4.1. We do not know a priori that the left side of (4.6) is injective for all but
finitely many (or even infinitely many) u0 , though this injectivity does follow a posteriori
from our proof that Eη (F (T )) has rank 1; the a priori difficulty is due to the fact that κ is
not separably closed (see Theorem 4.4). However, even if we did know such injectivity, it
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
19
would be useless because our rank bounds for Eu0 (κ0 (T )) are not good enough. The purpose
of considering (4.6) is precisely to circumvent our lack of information concerning the groups
Eu0 (κ0 (T )).
We shall now undertake the geometric part of the argument (injectivity of the right side
of (4.6) for all but finitely many u0 ). This will be deduced from a more general specialization
result for abelian varieties. Let us isolate the essential geometric properties of Eη before we
pass to an axiomatized setup with an abelian variety. Consider the surface S = P1κ × P1κ
with factors having respective coordinates u and T . By general “smearing out” principles,
Eη extends to an elliptic curve EV over a dense open V ⊆ S. (In fact, there is a unique
maximal such open V , containing all others, and the elliptic curve EV extending Eη over
this V is unique. This follows from a general lemma of Faltings [7, §2, Lemma 1], but we do
not need it.) Pick some choice of V and EV . There are finitely many (if any) codimension-1
points in S not in V , and if Eη has good reduction at such a point s then we can “smear
out” the proper Néron model over OS,s and glue it to EV so as to increase V to contain s.
Doing this finitely many times, we may assume V contains all codimension-1 points of S
where Eη has good reduction.
The complement S − V consists of finitely many curves and isolated closed points. Since
Eu0 is smooth for all closed points u0 ∈ A1κ , the curves in the complementary locus
S − V ⊆ P1κ × P1κ
are “non-vertical” except for possibly {∞} × P1κ . Put in geometric terms, when the bad
locus for Eη over S is fibered over the T -line it “moves” in the fibers St = P1 except for
possibly at the point ∞ in these fibers. We need to analyze the situation along the vertical
line u = ∞.
Lemma 4.2. The elliptic curve Eη in (4.1) has bad reduction at the codimension-1 generic
point η∞ of the line u = ∞ in S, with reduction type that is potentially good. The ramification of Eη [2] at η∞ is tame.
Proof. Since degu (h(T 2 + u)) = 2p, we see from (4.4) that degu (∆) = 18p is not divisible by
12. Therefore, there is bad reduction at η∞ . The j-invariant j(Eη ) is a unit at η∞ because j
in (2.5) is a unit at ∞, so the reduction at η∞ is potentially good. Since Eη [2](η∞ ) 6= O and
the residue characteristic at η∞ is not 2, the 2-torsion Eη [2] is tamely ramified at η∞ . Now we pass to a general situation that uses the properties proved in Lemma 4.2. Let
k be a separably closed field and let S be a connected geometrically-normal k-scheme of
finite type, equipped with a surjective k-morphism S → P1k whose fibers are geometrically
reduced and whose generic fiber is geometrically irreducible. By [10, IV3 , 9.7.7] there is a
dense open in P1k over which S has geometrically integral fibers. In the above discussion,
k = κ and S is the product of the projective u-line and projective T -line over k with
projection S → P1k onto the u-line.
Let A be an abelian variety of dimension g ≥ 1 over the function field k(S). For all but
finitely many closed points u ∈ P1k , A has good reduction Aηu at the codimension-1 generic
point ηu of the geometrically integral fiber Su in the normal S; we write k(Su ) to denote the
function field of this fiber. By the valuative criterion for properness we have a specialization
mapping
ρu : A(k(S)) → Aηu (k(Su ))
20
B. CONRAD, K. CONRAD, AND H. HELFGOTT
for such u. (Since we are not assuming that the Chow k(S)/k-trace of A vanishes, A(k(S))
might not be finitely generated. Hence, ρu cannot be defined by elementary denominatorchasing with a finite set of elements and their relations in A(k(S)), so we really do need the
valuative criterion for properness in order to define ρu0 ; more specifically we cannot expect
A(k(S)) to “smear out” beyond the codimension-1 local ring on S at the generic point ηu
of Su .) Motivated by the goal of proving that the right side of (4.6) is injective with only
finitely many exceptions, we want to analyze the kernel of the reduced map
ρu mod n : A(k(S))/n · A(k(S)) → Aηu (k(Su ))/n · Aηu (k(Su ))
for suitable integers n and for u avoiding a finite set of closed points on P1k . To this end, it
is convenient to first prove a general finiteness lemma.
Lemma 4.3. Let V be a geometrically integral variety over a field k and let B be an
abelian variety over K = k(V ). For all non-zero integers m with char(k) - m, the group
B(K)/m · B(K) is finite if k is separably closed. The same holds for arbitrary non-zero
integers m if k is algebraically closed.
Proof. We shall use Chow’s theory of the K/k-trace [18, Ch. VIII]. Here are the key points
of this theory (for our purposes). In the category of pairs (B0 , f0 ) consisting of an abelian
variety B0 over k and a map f0 : (B0 )K → B of abelian varieties over K, there is a final
object (TrK/k (B), τ ) and the canonical map τ : (TrK/k (B))K → B has infinitesimal kernel.
This object is the K/k-trace of B. Obviously the map
τ
TrK/k (B)(k) ,→ TrK/k (B)(K) → B(K)
is injective. The Lang–Néron theorem [17, Thm. 1] says that the quotient group
(4.7)
B(K)/TrK/k (B)(k)
is finitely generated. (To the best of our knowledge, all published references on these topics
are written in pre-Grothendieck terminology; the reader is referred to [5] for a discussion of
the Chow trace and Lang–Néron theorem using scheme-theoretic methods.)
Now assume that k is separably closed. Since TrK/k (B)(k) is the group of rational points
of an abelian variety over a separably closed field, it is m-divisible (and the restriction
char(k) - m can be removed if k is algebraically closed). Thus,
TrK/k (B)(k) ⊆ m · B(K),
so
B(K)/m · B(K) ' (B(K)/TrK/k (B)(k))/m · (B(K)/TrK/k (B)(k)).
This yields the desired finiteness because (4.7) is finitely generated.
We return to the abelian variety A/k(S) , described before Lemma 4.3.
Theorem 4.4. Assume that k is separably closed and that for all closed points u ∈ P1k
distinct from ∞, A has good reduction at some generic point of the (possibly reducible)
geometrically-reduced fiber Su . Assume moreover that at some generic point η∞ of the fiber
S∞ there is potentially good reduction.
Fix n ∈ Z with |n| > 1 such that char(k) - n, and assume that the Galois splitting field
of the finite étale k(S)-group A[n] is tamely ramified at the codimension-1 point η∞ ∈ S.
The mod-n reduction
ρu mod n : A(k(S))/n · A(k(S)) → Aηu (k(Su ))/n · Aηu (k(Su ))
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
21
of the specialization map along Su is injective for all but finitely many closed points u ∈ P1k .
The tameness assumption is equivalent to the condition that A acquires good reduction
over a finite separable extension of k(S) that is tame at a place over ηη (this is explained
in the proof), and so this hypothesis is automatically satisfied when every positive prime
` ≤ 2g + 1 is a unit in k (that is, char(k) = 0 or char(k) > 2g + 1). Thus, by setting g = 1
and n = 2 in Theorem 4.4, we may conclude via Lemma 4.2 (which also gives the desired
tameness in characteristic 3) that the right side of (4.6) is injective for all but finitely many
u0 ∈ A1κ (κ).
Proof. The hypotheses on S and A are preserved under extension of the base field. Moreover,
if k is an algebraic closure of k then we claim that the natural map
A(k(S))/n · A(k(S)) → A(k(S))/n · A(k(S))
is injective, so we may reduce to the case when k is algebraically closed. The case of
characteristic 0 is trivial, so we can assume char(k) = p > 0. It suffices to check more
generally that if K is a field with characteristic p > 0 and G is a commutative K-group
of finite type then the map G(K)/n · G(K) → G(K 0 )/n · G(K 0 ) is injective for any purely
inseparable algebraic extension K 0 /K and any integer n not divisible by p. We may assume
K 0 = K 1/p , so we get an identification G(K 0 ) ' G(p) (K) that identifies the inclusion
G(K) → G(K 0 ) with the map on K-points induced by the relative Frobenius morphism
FG : G → G(p) . Since [p] : G → G factors through FG [11, VIIA , §4.3], it suffices to prove
that the p-torsion in G(K)/n · G(K) vanishes, and this is clear since p - n.
Let W ⊆ S be a dense open such that A extends to an abelian scheme AW over W . The
complement S − W contains at most finitely many codimension-1 points of S, and if A has
good reduction at any such point s then we may glue AW with a smearing-out of the proper
Néron model of A over OS,s to increase W to contain a neighborhood of s. Thus, by the
hypothesis on reduction for A, we may suppose that no fiber Su over a closed any point
u ∈ P1k is disjoint from W except for possibly S∞ . This property of W is unaffected by
shrinking W in codimension ≥ 2. Let η be the generic point of S.
By Lemma 4.3 with V = S, A(k(S))/n·A(k(S)) is finite. We conclude from the pigeonhole
principle that if ρu mod n has nontrivial kernel for infinitely many u (ignoring the finitely
many for which Su is reducible, in which case ρu is not defined), then some non-zero
R ∈ A(k(S))/n · A(k(S))
is killed by ρu mod n for infinitely many u. Thus, it suffices to prove that if Rη ∈ A(k(S))
has the property that ρu (Rη ) lies in n·Aηu (k(Su )) for infinitely many u (ignoring the finitely
many u for which ρu is not defined) then Rη ∈ n · A(k(S)).
Choose Rη ∈ A(k(S)) such that ρu (Rη ) lies in n · Aηu (k(Su )) for infinitely many u.
By denominator-chasing, Rη extends (uniquely) to RU ∈ AW (U ) for some dense open
U ⊆ W . The valuative criterion for properness extends RU over each of the finitely many
codimension-1 points of W not contained in U . Thus, by shrinking W in codimension ≥ 2 if
necessary, we may assume that Rη extends to a section R ∈ AW (W ) of the abelian scheme
AW → W .
The pullback of [n] : AW → AW along R ∈ AW (W ) is a finite étale cover
(4.8)
[n]−1 (R) → W.
22
B. CONRAD, K. CONRAD, AND H. HELFGOTT
Our goal is to prove that (4.8) has a section over the generic point η = Spec k(S) of W . Let
L be a residue field on [n]−1 (R)η = [n]−1 (Rη ), so L is a finite separable extension of k(S),
say with degree dL . We want dL = 1 for some such L.
Lemma 4.5. For each L, the subfield k(P1 ) is algebraically closed in L.
Proof. Let K/k(P1 ) be the algebraic closure of k(P1 ) in L, so K/k(P1 ) is a finite separable
extension because L/k(P1 ) is a finitely generated separable extension (as k(S) is separable
over k(P1 ), since the generic fiber of S → P1 is geometrically integral). The intermediate
fields K and k(S) in the separable extension L/k(P1 ) are linearly disjoint over k(P1 ) because
K/k(P1 ) is algebraic and k(P1 ) is algebraically closed in k(S). Thus, if θ ∈ P1 is the generic
point then the function field
(4.9)
K(Sθ ) := K ⊗k(P1 ) k(S)
of the geometrically integral generic fiber Sθ/K is identified with the intermediate composite
field K · k(S) in L/k(S). The hypothesis on the good reduction of A implies that for every
closed point u ∈ P1 − {∞}, some generic point ηu of the reduced fiber Su lies in W . Hence,
since [n]−1 (R) → W is a finite étale cover, the residue field L on [n]−1 (Rη ) is unramified over
the discrete valuation on k(S) arising from some such ηu for every closed point u ∈ P1 −{∞}.
It follows that for every such u, the intermediate finite separable extension K(Sθ )/k(S) is
also unramified at some generic point ηu of Su .
We also need to understand the ramification behavior of L/k(S) at the discrete valuation
on k(S) arising from a generic point η∞ on the reduced fibral curve S∞ such that A has
potentially good reduction over a tame extension at η∞ ; the existence of such an η∞ was
one of our initial assumptions on A. We claim that L/k(S) is tamely ramified at all places
of L over η∞ . Some care will be required because L/k(S) may be non-Galois.
The first step is to check that L admits at least one place that is tame over η∞ , and
to do this it suffices to choose a separable closure of the residue field at η∞ and to show
that [n]−1 (Rη ) splits over a tame extension of the fraction field of the associated strict
sh . Since A[n] is assumed to be tamely ramified at η , there exists a
henselization OS,η
∞
∞
sh
finite tame extension F 0 over the fraction field of OS,η
such
that
A[n]
F 0 is a constant
∞
group. We can assume |n| > 1, so there exists a prime `|n and ` 6= char(k). Since A[n]F 0
is constant, the Galois-action on the `-adic Tate module of A/F 0 has pro-` image that is
finite (since A has potentially good reduction at η∞ ), so after replacing F 0 with a suitable
`-power (hence tame) extension we can assume that A/F 0 has good reduction. Let A denote
sh
the proper Néron model of A/F 0 over the integral closure OF 0 of OS,η
in F 0 . The group
∞
A(F 0 ) = A (OF 0 ) is n-divisible because OF 0 is strictly henselian and n is not divisible by
the residue characteristic of OF 0 , so [n]−1 (Rη )(F 0 ) 6= ∅. Since A[n]F 0 is split, it follows
that the étale A[n]-torsor [n]−1 (Rη ) must therefore be split over F 0 . Hence, L admits a
k(S)-embedding into F 0 , so L/k(S) is tamely ramified at some place wL over the discrete
valuation on k(S) arising from η∞ .
By definition, L is a residue field on an étale A[n]-torsor [n]−1 (Rη ) over k(S), and (by
hypothesis) the k(S)-group A[n] splits over a finite Galois extension M/k(S) that is tamely
ramified over η∞ . Thus, the factor fields of the finite étale M -algebra L ⊗k(S) M are residue
fields on the torsor [n]−1 (Rη )M for a finite constant group over M (namely, the constant
group A[n]M ). Hence, the factor fields Li of L ⊗k(S) M are Galois over M and the Li ’s are
pairwise M -isomorphic. Pick a place wM on M lifting the place η∞ on k(S). Since wM
and wL lift the same place on k(S), we can find a factor field Liw of L ⊗k(S) M and a place
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
23
viw on Liw that lifts the places wL and wM . The place viw on Liw must be tame over the
place wM because wL is tame over η∞ on k(S). The extension Liw /M is Galois, so Liw /M
is tame at all places over wM . Since the Li ’s are pairwise M -isomorphic and wM is an
arbitrary place on M over η∞ , every Li is tame over every place on M lifting η∞ . Since all
places of M over η∞ are tame over η∞ , we conclude that all places lying over η∞ on each
Li are tame over η∞ . Upon choosing some Li0 , the extension L/k(S) is a subextension of
Li0 /k(S) and hence L/k(S) is tamely ramified at all places over η∞ . The same therefore
holds for the intermediate extension K(Sθ )/k(S) in (4.9).
Summarizing our conclusions, the finite separable extension K(Sθ ) = K ⊗k(P1 ) k(S) over
k(S) is unramified at some generic point of the reduced fiber Su for each u 6= ∞ and is
tamely ramified over some generic point of the reduced fiber S∞ . The reducedness of the
fibers implies that a uniformizer at a closed point u ∈ P1 pulls back to be a uniformizer in
the local ring at the codimension-1 point ηu on the normal surface S. Hence, the discrete
valuation on k(P1 ) associated to u has ramification index 1 under the discrete valuation on
k(S) associated to ηu , and the corresponding residue field extension is separable (because
the residue field at u is the field k that is algebraically closed). It follows by classical
valuation theory and (4.9) that if ηu is unramified (resp. tamely ramified) in K(Sθ ) then
u must be unramified (resp. tamely ramified) in K. Hence, the finite separable (possibly
non-Galois) extension K/k(P1 ) is unramified away from ∞ and is tamely ramified at all
places over ∞. Since k is separably closed, we conclude that K = k(P1 ).
We return to the proof of Theorem 4.4. Let CL be the connected component of [n]−1 (R)
with function field L. Since L/k(P1 ) is a finitely generated separable extension with transcendence degree 1, it follows from Lemma 4.5 that the fiber of CL over the generic point
of P1k must be geometrically integral over k(P1 ). Hence, by [10, IV3 , 9.7.7], there is a
Zariski-dense open UL ⊆ A1k such that the fiber (CL )u is geometrically integral over k(u)
for all u ∈ UL . By removing finitely many closed points from UL , we may (and do) also
assume that Su is geometrically integral over k(u) for all u ∈ UL . Since [n]−1 (R) is finite
étale over W and the open subset Wu ⊆ Su is non-empty for all u ∈ A1k , the finite étale
map (CL )u → Wu has degree
[k(CL ) : k(W )] = [L : k(S)] = dL
for all points u ∈ UL .
Choose u ∈ ∩L UL , where L runs over all the residue fields on [n]−1 (Rη ). We have just
proved that the fiber (CL )u is connected (even geometrically integral over k(u)) for all L.
It follows that {(CL )u }L is the set of connected components of the finite étale Wu -scheme
[n]−1 (R)u = [n]−1 (Ru ) and the map
(CL )u → Wu ⊆ Su
is étale with generic degree dL for all L. Membership in ∩L UL omits only finitely many
closed points u, so by the hypothesis ρu (Rη ) ∈ n · Aηu (k(Su )) for infinitely many u (with
ρu (Rη ) the generic point of the section Ru of AW over Wu ) we conclude that there exists a
closed point u0 ∈ ∩L UL such that
ρu0 (Rη ) ∈ n · Aηu0 (k(Su0 )).
In particular, the Wu0 -étale scheme [n]−1 (Ru0 ) has a k(Su0 )-rational point. This rational
point lies in some fibral connected component (CL0 )u0 , so the generic degree dL0 of this
component over Su0 must equal 1.
24
B. CONRAD, K. CONRAD, AND H. HELFGOTT
5. Generic rank bound II. Arithmetic arguments
Our remaining task is to prove that the bottom side of (4.6) is injective for infinitely
many closed points u0 ∈ A1κ . In Theorem 5.1 we will find infinitely many points u0 such
that the elliptic curve Eu0 /κ0 (T ) over the global field κ0 (T ) has exactly two places of bad
reduction, and in §6 we will prove injectivity along the bottom of (4.6) for such u0 .
As preparation for the study of the image along the bottom side of (4.6) for well-chosen
closed points u0 ∈ A1κ , we fix an arbitrary u0 and find the reduction type of Eu0 at each
place of κ0 (T ). After we find these reduction types, the points u0 that will become our
focus of interest will be those such that Eu0 has the smallest possible number of physical
points of bad reduction on the T -line P1κ0 .
Recall that (4.1) defines Eη in terms of h(T 2 + u), where h(T ) = cT 2p + du. From (4.4),
the discriminant of (4.1) involves h(T 2 + u) and 1 + 4h(T 2 + u). Under a u0 -specialization,
h(T 2 + u) becomes a pth power in κ0 [T ]:
1/p
h(T 2 + u)|u=u0 = (c1/p (T 2 + u0 )2 + d1/p u0 )p .
Likewise, 1 + 4h(T 2 + u) specializes to a pth power in κ0 [T ]:
1/p
(1 + 4h(T 2 + u))|u=u0 = (1 + 4(c1/p (T 2 + u0 )2 + d1/p u0 ))p .
For all but finitely many closed points u0 ∈ A1κ , the pth-root polynomials
1/p
π1 := c1/p (T 2 + u0 )2 + d1/p u0 , π2 := 1 + 4π1
(5.1)
are separable in κ0 [T ]. (These quartics over the finite field κ0 may be reducible for many
points u0 , and so even in characteristic p > 3 these quartics may be fail to be separable
for some non-empty finite set of points u0 . In Theorem 5.1 below, we will show that for
infinitely many u0 we can do much better than mere separability.) We now restrict attention
to those u0 such that the two polynomials in (5.1) are both separable. (Our notation π1
and π2 does not indicate the dependence on u0 ; it would be more accurate to write π1,u0
and π2,u0 , but we simply ask the reader to remember the dependence on u0 .)
Specializing (4.4) at u0 , the Weierstrass model that defines Eu0 /κ0 (T ) has parameters
(5.2)
∆|u=u0 = 16π18p π2p ,
c4 |u=u0 = 16π12p (1 + 3π1p ) = 16π12p (π2 − π1 )p ,
and this Weierstrass model is integral away from T = ∞. Thus, the only possible bad
reduction for Eu0 over the T -line P1κ0 is at ∞ and at the zeros of π1 and π2 .
What is the behavior of Eu0 at the point ∞ ∈ P1κ0 ? We return to Lemmas 2.1 and 2.2.
Both π1 and π2 have degree 4 in κ0 [T ], so by (5.2) we have
ord∞ (∆|u=u0 ) = −36p.
When char(κ) > 3,
ord∞ (c4 |u=u0 ) = −12p, ord∞ (j|u=u0 ) = 0.
When char(κ) = 3,
ord∞ (c4 |u=u0 ) = −8p, ord∞ (j|u=u0 ) = 12p.
Thus, there is potentially good reduction at T = ∞ in all cases, and Lemma 2.1 ensures
that this reduction is good.
Now we analyze the reduction types at points xj in the zero-scheme of πj on P1κ0 . Since
π1 is separable in κ0 [T ] (by our choice of u0 ), we see from (5.2) that for any x1 ,
ordx1 (∆|u=u0 ) = 8p,
ordx1 (c4 |u=u0 ) = 2p ≡ 2 mod 4.
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
25
Therefore ordx1 (j(Eu0 )) = 6p − 8p = −2p < 0. By Lemma 2.2, there must be (potentially
multiplicative) additive reduction at x1 . Similarly, we compute
ordx2 (∆|u=u0 ) = p,
ordx2 (c4 |u=u0 ) = 0,
so ordx2 (j(Eu0 )) = −p < 0. By Lemma 2.2, there is multiplicative reduction at x2 .
We have shown that the Néron model N (Eu0 ) → P1κ0 enjoys the following reduction
properties:
(a) good reduction at all closed points of P1κ0 away from zeros of π1 and π2 ,
(b) multiplicative reduction at zeros x2 of π2 , with ordx2 (ju0 ) = −p.
(c) additive reduction at zeros x1 of π1 , with ordx1 (ju0 ) = −2p < 0.
Properties (b) and (c) will be used in our work with Néron models and Selmer groups in §6,
but now we focus on (a). The most favorable u0 ’s for our purposes will be those such that
Eu0 has the least possible number of physical points of bad reduction, so we want to find
many u0 such that π1 and π2 are both irreducible in κ0 [T ]. For such u0 , Eu0 has exactly
two physical points of bad reduction on P1κ0 .
Theorem 5.1. There exist infinitely many closed points u0 ∈ A1κ such that π1 , π2 ∈ κ0 [T ]
are irreducible.
To find the infinitely many u0 as in the theorem will require some effort, so let us first
sketch the basic idea. In (5.1) we see that u0 ∈ κ0 intervenes in π1 and π2 through the
1/p
value u0 ∈ κ0 , so to put ourselves in the position of specializing polynomials in u we apply
arithmetic Frobenius of κ0 to the coefficients of π1 and π2 . This leads us to consider the
polynomials
(5.3)
Π1 (u, T ) := c(T 2 + up )2 + du, Π2 (u, T ) := 1 + 4Π1 (u, T ) ∈ κ[u][T ].
For any closed point u0 ∈ A1κ , the specialization Πj (u0 , T ) ∈ κ0 [T ] is the image of πj ∈ κ0 [T ]
under the arithmetic Frobenius automorphism of κ0 . Thus, Theorem 5.1 is equivalent to the
existence of infinitely many u0 ∈ A1κ such that both Π1 (u0 , T ) and Π2 (u0 , T ) are irreducible
in κ0 [T ], where κ0 = κ(u0 ) is varying with u0 . It is this equivalent statement that we will
actually prove (Theorem 5.8 below).
Expanding Π1 and Π2 as polynomials in κ(u)[T ], we have
du
du
1
4
p 2
2p
4
p 2
2p
(5.4)
Π1 = c T + 2u T + u +
, Π2 = 4c T + 2u T + u +
+
.
c
c
4c
It is left to the reader to check that Π1 and Π2 are separable and irreducible over κ(u), via
the following elementary criterion concerning polynomials of the form X 4 + aX 2 + b.
Lemma 5.2. Let K be a field with char(K) 6= 2. A polynomial f = X 4 + aX 2 + b ∈ K[X]
is separable if and only if b and a2 − 4b are non-zero. It is irreducible if b and a2 − 4b are
non-squares in K × .
Proof. The condition for separability is obvious. We now assume that b and a2 − 4b are
non-squares in K × . Since a2 − 4b is not a square, f has no roots in K and has no factors
of the form X 2 − c in K[X]. Thus, if we write the four roots of f in a splitting field as ±r1
and ±r2 , a non-trivial monic factor of f in K[X] must have the form (X ± r1 )(X ± r2 ). If
such a factor exists then r1 r2 ∈ K and b = (r1 r2 )2 , contradicting the assumption that b is
a non-square in K.
26
B. CONRAD, K. CONRAD, AND H. HELFGOTT
In view of the irreducibility of each Πj in κ(u)[T ] and our desire to prove
Π1 (u0 , T ), Π2 (u0 , T ) ∈ κ0 [T ]
are irreducible for infinitely many closed points u0 ∈ A1κ , our problem resembles Hilbert
irreducibility. However, finite fields are not Hilbertian and anyway we are not generally
specializing u at elements of κ (since [κ0 : κ] > 1 with only finitely many exceptions).
The main idea that will produce the desired u0 ’s is the following theorem. It gives a
group-theoretic criterion for a polynomial over a global field to specialize to an irreducible
polynomial over the residue field at infinitely many places (see Remark 5.4).
Theorem 5.3. Let K be a global field and let f ∈ K[T ] be a monic separable irreducible
polynomial of degree n. Let K 0 /K be a splitting field for f and let G = Gal(K 0 /K). For any
non-archimedean place v of K at which f has integral coefficients, f mod v is irreducible
over the residue field Fv at v if and only if v is unramified in K 0 and the Frobenius elements
over v in G act as n-cycles on the set of roots of f in K 0 .
Proof. Let r be a root of f in K 0 . If f is v-integral and f mod v is separable, then the
discriminant of f is a v-adic unit, so v is unramified in K(r). Since K 0 is a composite of
such extensions of K, in such cases v must be unramified in K 0 . Let v 0 be a place of K 0
over a place v in K that is unramified in K 0 . The action of Frob(v 0 |v) on the n roots of f
in K 0 is identified with the action of the finite-field Frobenius x 7→ x#Fv on the full set of
n roots of f mod v (in Fv0 ). In particular, f mod v is irreducible over Fv if and only if v is
unramified in K 0 and Frob(v 0 |v) acts as an n-cycle on the roots of f .
Remark 5.4. In the setting of Theorem 5.3, if r ∈ K 0 is a root of f and H ⊆ G is the
subgroup associated to the intermediate field K(r) ⊆ K 0 , then an element γ ∈ G acts as
an n-cycle on the set of roots of f in K 0 if and only if the cyclic subgroup hγi is a set of
representatives for the coset space G/H of order n. We conclude by Chebotarev’s density
theorem that f mod v is irreducible for infinitely many places v of K if and only if G/H
admits a set of representatives that is a cyclic subgroup of G.
Corollary 5.5. Let K be a global field and let f ∈ K[T ] be a monic separable irreducible polynomial of degree n. The following are equivalent (restricting attention to nonarchimedean places at which the coefficients of f are integral):
(1) There is some place v such that f mod v is irreducible.
(2) There is a positive proportion of places v such that f mod v is irreducible.
Proof. The implication (2) ⇒ (1) is trivial, and the converse follows from Theorem 5.3 and
Chebotarev’s density theorem.
Example 5.6. Let f satisfy the hypotheses in Theorem 5.3, and let {r1 , . . . , rn } be an
ordering of the set of roots of f in K 0 . Identify G = Gal(K 0 /K) with a subgroup G ⊆ Sn
via the G-action on the rj ’s. By Theorem 5.3, f mod v is irreducible for infinitely many v
if and only if G contains an n-cycle.
(1) If G is isomorphic to Sn as abstract groups (where n = deg f ), then G = Sn . Since
G contains an n-cycle, f mod v is irreducible for infinitely many v.
(2) What if G is isomorphic to An (as abstract groups)? Since An embeds into Sn with
only one possible image, and An contains an n-cycle if and only if n is odd, we see
that f mod v is irreducible infinitely often if and only if n is odd.
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
27
(3) What if G (and thus G) is isomorphic to Dn (as abstract groups) with n > 2? Then
G is isomorphic to Dn as a permutation group, so f mod v is irreducible infinitely
often.
The identification of G with Dn as a permutation group was explained to us by
D. Pollack. Write G = hσ, τ i, where σ n = 1, τ 2 = 1 and τ στ −1 = σ −1 . Since hσi is
normal in G and G is a transitive subgroup of Sn , all hσi-orbits have the same length.
Therefore, since σ has order n it must be an n-cycle. Writing σ = (1, 2, . . . , n), the
condition τ στ −1 = σ −1 says (τ (1), τ (2), . . . , τ (n)) = (n, n − 1, . . . , 1) as n-cycles.
We can replace τ in the presentation of G with τ σ k for any k, so we may assume
τ (1) = 1. Identifying j with e2πi(j−1)/n , σ and τ are now the standard generators
for Dn in its natural action on an n-gon.
(4) What if f is a normal polynomial; i.e., G has order n? A transitive subgroup of
order n in Sn contains an n-cycle if and only if it is cyclic, so the reduction of f at
infinitely many places is irreducible if G is cyclic but not otherwise.
(5) In the preceding four examples, the structure of the Galois group G as an abstract
group was sufficient to determine if the permutation group G contains an n-cycle.
However, this is not generally the case. For example, there is a group of size 2592 =
25 · 34 admitting two transitive actions of degree 12 such that one action contains
12-cycles and the other does not. The actions were found for us by N. Boston using
MAGMA; they are the 245th and 246th transitive groups of degree 12 in MAGMA’s
enumeration. MAGMA also realizes both of these transitive groups as Galois groups
over Q.
We now apply these ideas to the polynomials Π1 (u, T ) and Π2 (u, T ) from (5.4). To
determine their Galois groups over κ(u), we use the following classical lemma.
Lemma 5.7. Let K be a field with char(K) 6= 2, and let f = X 4 + aX 2 + b ∈ K[X] be
separable and irreducible. Let K 0 /K be a splitting field and G = Gal(K 0 /K). We have the
following possibilities for G as an abstract group:
2
×
• G ' Z/4Z if √
and only if
√ b(a − 4b) ∈ K is a square, in which case the quadratic
subfield is K( b) = K( a2 − 4b),
• G ' Z/2Z × Z/2Z if and onlypif b ∈ K × is a square,
case the quadratic
p in which
√
√
√
2
subfields
are K( a − 4b) , K( −a + 2 b), and K( −a − 2 b) for a fixed choice
√
of b ∈ K × ,
×
• G ' D4 if and only if b and√ b(a2 − 4b) are
p in K , in which case
√ not squares
2
2
the quadratic subfields are K( a − 4b), K( b), and K( b(a
p − 4b)). The unique
0
quadratic subfield over which K is a cyclic extension is K( b(a2 − 4b)).
Proof. This classification of Galois groups according to properties of the coefficients can be
found as an exercise in many basic algebra books, although usually it is stated only over
Q. In that spirit, the other assertions are left as an exercise for the reader.
To apply Lemma 5.7 to Π1 and Π2 , we look at (5.4) and label the coefficients inside the
parentheses as
du
du
1
A1 = 2up , B1 = u2p +
, A2 = 2up , B2 = u2p +
+ ,
c
c
4c
4
2
×
so Πj = T + Aj T + Bj modulo κ -scaling. Since we used Lemma 5.2 to prove that each
Πj is separable and irreducible, we already know that Bj and A2j − 4Bj are non-squares
28
B. CONRAD, K. CONRAD, AND H. HELFGOTT
in κ(u)× . A direct calculation also shows that Bj (A2j − 4Bj ) is a non-square in κ(u)× .
Therefore, by Lemma 5.7, each of Π1 and Π2 has Galois group over κ(u) that is isomorphic
to D4 . Example 5.6(3) now tells us that Π1 and Π2 each have infinitely many irreducible
u0 -specializations. What about simultaneous irreducible specializations? This is what we
need to resolve in order to complete the proof of Theorem 5.1.
Theorem 5.8. There exist infinitely many u0 such that Π1 (u0 , T ) and Π2 (u0 , T ) are both
irreducible in κ0 [T ].
Proof. Let Lj /κ(u) be a splitting field of Πj , so Gal(Lj /κ(u)) is isomorphic to D4 . We will
show L1 and L2 are linearly disjoint over κ(u). It will then follow, by the Chebotarev density
theorem, that any pair of Frobenius elements in Gal(L1 /κ(u)) × Gal(L2 /κ(u)) are both
attached to infinitely many common places on κ(u). Theorem 5.3 and Example 5.6(3) then
imply there are infinitely many u0 such that Π1 (u0 , T ) and Π2 (u0 , T ) are both irreducible
in κ0 [T ].
Any intermediate extension in Lj /κ(u), other than κ(u), contains a quadratic extension
of κ(u) since every proper subgroup of a 2-group is contained in a subgroup of index 2.
We will show that L1 and L2 do not contain quadratic subfields (over κ(u)) that are κ(u)isomorphic, so they must be linearly disjoint over κ(u).
Inspection shows the only occurrences of non-trivial common factors among
(5.5)
B1 , A21 − 4B1 , B2 , A22 − 4B2
are: the linear polynomial A21 − 4B1 divides B1 and (when c = 4d2p ) the linear polynomial
A22 − 4B2 divides B1 . Since B1 is separable with deg B1 > 2, we conclude that the four
elements in (5.5) are multiplicatively independent modulo squares in κ(u)× . This independence modulo squares, coupled with the list of quadratic subfields in the D4 -case of Lemma
5.7, shows L1 and L2 do not share a common quadratic extension of κ(u). Thus, they are
linearly disjoint over κ(u).
6. Generic rank bound III. Cohomological arguments
By Theorem 5.1, there are infinitely many closed points u0 ∈ A1κ such that the “specialized” polynomials
(6.1)
1/p
π1 = c1/p (T 2 − u0 )2 + d1/p u0 , π2 = 1 + 4π1
in κ0 [T ] are both irreducible. These are the only u0 that we shall henceforth consider.
We view π1 and π2 as closed points on A1κ0 ⊆ P1κ0 . The arithmetic of
(6.2)
Eu0 : y 2 = x3 + π1p x2 − π13p x
for such u0 is our focus of interest, as this will provide the information that we need to prove
that the image of the bottom map in (4.6) has dimension ≤ 2 for these points u0 . This will
complete the proof that Eη (F (T )) has rank 1, thereby concluding the proof of Theorem 1.1.
Rather than work with Eu0 , it will simplify matters to work with the elliptic curve
(6.3)
Eu0 0 : y 2 = x3 + π1 x2 − π13 x;
this elliptic curve is p-isogenous to Eu0 = (Eu0 0 )(p) , so by oddness of p it follows that the
map along the bottom of (4.6) is canonically identified with the map
(6.4)
Eu0 0 (κ0 (T ))/2 · Eu0 0 (κ0 (T )) → Eu0 0 (κ0 (T ))/2 · Eu0 0 (κ0 (T )),
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
29
where κ0 is an algebraic closure of κ0 . We shall prove that (6.4) is injective for the points
u0 presently under consideration.
The reduction properties of the Néron model N (Eu0 ) → P1κ0 were worked out in §5 (see
above Theorem 5.1), and the additive and multiplicative properties are the same for the
Néron model of the isogenous elliptic curve Eu0 0 . Thus, letting ju0 0 = j(Eu0 0 ), for points u0
such that π1 and π2 are irreducible in κ0 [T ] we obtain the following properties for N (Eu0 0 ):
• good reduction at all closed points of P1κ0 away from π1 and π2 ,
• multiplicative reduction at π2 , with ordπ2 (ju0 0 ) = −1.
• additive reduction at π1 that is potentially multiplicative, with ordπ1 (ju0 0 ) = −2.
By the theory of Tate models for multiplicative reduction, the component group for the
Néron model at π2 is trivial, so the π2 -fiber N (Eu0 0 )π2 is a torus.
Fix a geometric point π 1 over the point {π1 } ∈ P1κ0 . The reduction at π1 is (additive
and) potentially multiplicative, and ordπ1 (ju0 0 ) = −2 is negative and even. We need to know
the structure of the component group of the additive geometric fiber of the Néron model at
π 1 . This can be deduced from Tate’s algorithm, but we give here a direct proof via general
principles.
Lemma 6.1. Let R be a discrete valuation ring with residue field k and fraction field K, and
let E be an elliptic curve over K with Néron model N (E) over R. Assume that ordR (j(E))
is negative and even, that E has additive reduction over R, and that char(k) 6= 2.
If k is perfect and k/k is an algebraic closure, then the geometric component group
N (E)k /N (E)0k is isomorphic to Z/2Z × Z/2Z.
Proof. The formation of the Néron model over a discrete valuation ring commutes with
base change to a strict henselization and to a completion, so we may assume that k is
separably closed and that R is complete. Since ordR (j(E)) < 0, it follows from Tate’s
theory that E is a quadratic twist of the Tate curve E0 over K with j-invariant j(E). Since
char(k) 6= 2 and R is strictly henselian, the Tate parameter qE0 must be a square in K ×
because ordR (qE0 ) = − ordR (j(E0 )) = − ordR (j(E)) is even. Thus, the 2-torsion on the
Tate curve E0 is a constant group over K. This property of the 2-torsion is unaffected by
quadratic twisting, so E[2] is a constant group over K.
Using the Néron mapping property, we obtain a map of R-groups
Z/2Z × Z/2Z → N (E),
and by passing to k-fibers we arrive at a map of finite étale groups
Z/2Z × Z/2Z → N (E)k /N (E)0k .
This map is injective by the hypothesis that the reduction is additive and char(k) 6= 2. It
is a general fact that for any discrete valuation ring R and any elliptic curve E over the
fraction field of R, the component group of the closed fiber of the Néron model N (E) has
order at most 4 when E has additive reduction and the residue field k is perfect. This follows
from the relationship between N (E) and the minimal regular proper model E reg over R,
together with the combinatorial classification of the extended Dynkin diagrams that describe
the special fiber Ekreg (equipped with its intersection form) when k is algebraically closed;
see [19, 10.2].
Theorem 6.2. The 2-torsion subgroup N (Eu0 0 )[2] is quasi-finite, étale, and separated over
P1κ0 . It is finite étale of order 4 over P1κ0 − {π2 } and has fiber of order 2 over {π2 }.
30
B. CONRAD, K. CONRAD, AND H. HELFGOTT
Proof. Since all points on P1κ0 have residue characteristic not equal to 2, doubling on N (Eu0 0 )
is an étale map that has fiberwise-finite kernel. Hence, N (Eu0 0 )[2] is a quasi-finite, étale,
and separated P1κ0 -group, so it is finite over an open U ⊆ P1κ0 if and only if its fiber rank
is constant on U . Since N (Eu0 0 )π2 is a torus, N (Eu0 0 )[2]π2 = N (Eu0 0 )π2 [2] has order 2. For
x ∈ P1κ0 − {π1 , π2 } the fiber N (Eu0 0 )x is an elliptic curve, so its 2-torsion subgroup has order
4. It remains to check that N (Eu0 0 )[2]π1 has order 4. Consider the exact sequence of smooth
groups
(6.5)
0 → N (Eu0 0 )0π1 → N (Eu0 0 )π1 → N (Eu0 0 )π1 /N (Eu0 0 )π0 1 → 0.
By Lemma 6.1, the final term has order 4 and is killed by 2. Since we are not in characteristic
2, doubling is an automorphism of the additive group N (Eu0 0 )π0 1 , so (6.5) splits. This gives
the result.
Consider the two points P00 = (0, 0) and Q00 = (−π1 , π12 ) in Eu0 0 (κ0 (T )), where P00 is a
rational point of order 2 and (Q00 )(p) ∈ (Eu0 0 )(p) (κ0 (T )) = Eu0 (κ0 (T )) is the u0 -specialization
of (4.2).
Theorem 6.3. The natural map
(6.6)
Eu0 0 (κ0 (T )) = N (Eu0 0 )(P1κ0 ) → N (Eu0 0 )π1 /N (Eu0 0 )π0 1 ' Z/2Z × Z/2Z,
carries P00 and Q00 to linearly independent elements. In particular, the component group at
π1 is a constant group generated by the classes of P00 and Q00 .
Proof. The meaning of the theorem is that P00 and Q00 reduce into distinct non-identity
components of N (Eu0 0 )π1 . By [19, 9.4/35,37] and [19, 10.2/14], the smooth locus in a minimal
Weierstrass model is the relative identity component of the Néron model over any discrete
valuation ring. The Weierstrass model (6.3) is minimal at π1 . Since P00 and Q00 reduce to
the unique non-smooth point (0, 0) on the closed fiber of this model, we conclude that the
reductions of P00 and Q00 in the Néron model at π 1 do not lie in the identity component.
To see that the reductions of P00 and Q00 in the Néron model at π 1 lie in distinct components, we just have to check that the difference P00 − Q00 = −(P00 + Q00 ) also has reduction
not in the identity component on the π 1 -fiber of the Néron model; that is, the point P00 + Q00
should have reduction (0, 0) with respect to the minimal Weierstrass model (6.3) at π1 . It
is trivial to compute P00 + Q00 = (π12 , π13 ), and this has reduction (0, 0).
Theorem 6.4. Let κ0 be an algebraic closure of κ0 = κ(u0 ), with u0 ∈ A1κ a closed point
such that π1 and π2 as in (6.1) are irreducible in κ0 [T ]. The image of the canonical map
c : Eu0 0 (κ0 (T ))/2 · Eu0 0 (κ0 (T )) → Eu0 0 (κ0 (T ))/2 · Eu0 0 (κ0 (T ))
in (6.4) is spanned by c(P00 ) and c(Q00 ), so dimF2 image(c) ≤ 2.
Proof. Let δ : Eu0 0 (κ0 (T ))/2Eu0 0 (κ0 (T )) → S [2] (Eu0 0 /κ0 (T ) ) be the injective Kummer map to
h
the 2-torsion Selmer group. Let Kxh denote the fraction field of the henselization OP
1 ,x
S [2] (Eu0 0 /κ0 (T ) ),
P1κ0 .
κ0
of the local ring at a closed point x ∈
For any element σ ∈
the local
restriction
σx ∈ H1 (Kxh , Eu0 0 [2])
is in the image of the local Kummer map δx at x. Write σx = δx (ξx ) for a point
h
ξx ∈ Eu0 0 (Kxh ) = N (Eu0 0 )(OP
1
κ
0
,x ).
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
31
By Theorem 6.3, subtracting a suitable Z-linear combination of δ(P00 ) and δ(Q00 ) from σ gives
a Selmer class σ 0 such that σπ0 1 = δπ1 (ξπ0 1 ), where ξπ0 1 reduces into the identity component at
π1 . Thus, S [2] (Eu0 0 /κ0 (T ) ) is generated by δ(P00 ), δ(Q00 ), and classes σ 0 such that σπ0 1 = δπ1 (ξ 0 )
for some local point ξ 0 in Eu0 0 (Kπh1 ) that reduces into the identity component at π1 ; note
that this local property of σ 0 at π1 is independent of the non-canonical choice of ξ 0 since
any two choices differ by an element in [2](Eu0 0 (Kπh1 )) and doubling on N (Eu0 0 )π1 kills the
component group (by Lemma 6.1).
The doubling map on N (Eu0 0 ) is fiberwise surjective over P1κ0 away from {π1 } and doubling
is surjective on the additive identity component at π1 (since p 6= 2). Thus, for Selmer
classes σ 0 as above with local restriction σx0 = δx (ξx0 ), the image of ξx0 in Eu0 0 (Kxsh ) lies in
[2]Eu0 0 (Kxsh ) for every closed point x ∈ P1κ0 and every choice of ξx0 (with Kxsh denoting a
maximal unramified extension of Kxh ). In other words, the inertial restriction σx0 |Kxsh is a
trivial cohomology class for all x. Hence, S [2] (Eu0 0 /κ0 (T ) ) is spanned by the images of P00 and
Q00 and the intersection of this Selmer group with the subgroup of everywhere unramified
classes in H1 (κ0 (T ), Eu0 0 [2]).
Let us now recall how to describe the group of everywhere unramified classes in terms of
étale cohomology. Let G = N (Eu0 0 )[2] and P = P1κ0 , so G is a quasi-finite separated étale
commutative P-group. If we let iη : η → P be the canonical map from the generic point η
of P, then the identity N (Eu0 0 ) = iη∗ (Eu0 0 ) on the smooth site over P implies G = iη∗ (Gη )
as étale sheaves (by passing to 2-torsion subsheaves). Thus, using the étale topology, the
r
s
r+s (η, G ) has Er,0 = Hr (P, G), so
Leray spectral sequence Er,s
η
2
2 = H (P, R iη∗ (Gη )) ⇒ H
we get an exact sequence of low-degree terms
⊕βx M 0
α
(6.7)
0 → H1 (P, G) → H1 (η, Gη ) →
H (κ0 (x), H1 (Kxsh , G)).
x
Here α is the canonical restriction map to the generic point and βx is the canonical local
restriction map at the non-generic point x of P. Hence, H1 (P, G) ⊆ H1 (η, Gη ) is the group
of everywhere unramified classes.
In view of the preceding considerations, to prove Theorem 6.4 it suffices to prove that
the restriction map
H1 (κ0 (T ), Eu0 0 [2]) → H1 (κ0 (T ), Eu0 0 [2])
kills the subgroup H1 (P, G) of everywhere unramified classes, where G = N (Eu0 0 )[2]. We
will prove the stronger assertion that the map H1 (P, G) → H1 (Pκ0 , G) vanishes.
Let U 0 = P−{π2 } and let j 0 : U 0 ,→ P be the canonical open immersion. By Theorem 6.2,
G|U 0 is finite étale over U 0 and Gπ2 has order 2 over κ(π2 ). Thus, the nontrivial 2-torsion
point (0, 0) defines a short exact sequence of étale sheaves
(6.8)
0 → Z/2Z → G → j!0 (Z/2Z) → 0
over P. By considering the exact sequence of pullback sheaves on Pκ0 = P1κ0 and using the
vanishing of H1 (P1κ0 , Z/2Z), we arrive at a commutative square
(6.9)
H1 (Pκ0 , G)
O
/ H1 (Pκ , j 0 (Z/2Z))
0
O!
H1 (P, G)
/ H1 (P, j 0 (Z/2Z))
!
32
B. CONRAD, K. CONRAD, AND H. HELFGOTT
whose top side is injective and whose vertical maps are the natural pullback maps. It
therefore suffices to prove that the pullback map on the right side vanishes, and this is a
property that does not involve G.
Since j!0 (Z/2Z) is represented by an étale P-group that is quasi-affine over P (it is
the complement in (Z/2Z)P of the non-identity point over {π2 } ∈ P), the elements of
H1 (P, j!0 (Z/2Z)) are in bijection with isomorphism classes of (representable) étale j!0 (Z/2Z)torsors on P; the same holds over Pκ0 , and the right side of (6.9) is thereby identified with
base-change on torsors. Thus, we just need to prove that every étale j!0 (Z/2Z)-torsor on P
has a Pκ0 -point.
Let i : Spec κ0 (π2 ) ,→ P be the closed complement to U 0 , so we have a short exact
sequence
(6.10)
0 → j!0 (Z/2Z) → Z/2Z → i∗ (Z/2Z) → 0
of étale sheaves on P, with i∗ (Z/2Z) supported at one physical point on P. Thus, the
natural map
H1 (P, j!0 (Z/2Z)) → H1 (P, Z/2Z)
is injective. Since H1 (P, Z/2Z) = H1 (P1κ0 , Z/2Z) = H1 (κ0 , Z/2Z), clearly H1 (P, Z/2Z) has
order 2 with its nontrivial element represented by the nontrivial Z/2Z-torsor Pκ00 → P for
a quadratic extension κ00 /κ0 , and the subgroup H1 (P, j!0 (Z/2Z)) has order 1 or 2.
The fiber of the Z/2Z-torsor Pκ00 over the point π2 ∈ P is a split double cover of
Spec κ0 (π2 ) since degκ0 π2 = 4 and [κ00 : κ0 ] = 2. Removing one of the two points over
{π2 } in Pκ00 gives an open T ⊆ Pκ00 that is a nontrivial j!0 (Z/2Z)-torsor over P. Hence,
H1 (P, j!0 (Z/2Z)) has order 2 and contains T as its unique nontrivial element. Since T
obviously acquires a section upon extending the ground field κ0 to κ00 , we conclude that
T (Pκ0 ) is nonempty.
7. Nagao’s function field conjecture
We have already shown that, under the parity conjecture, (1.4) has “unexpected” rank
behavior for its specializations in T . A conjecture of Nagao [22, p. 14], to be reviewed below,
predicts that the Mordell–Weil rank of a non-constant elliptic curve over Q(T ) is a certain
limit of averages over mod p specializations as p runs over the primes. Nagao’s conjecture
admits a natural variant for elliptic curves over F (T ) for any global field F , and so in view of
the unusual behavior of Mordell–Weil ranks in (1.4) discussed in this paper one may be led
to wonder if it is necessary to introduce restrictions in the analogue of Nagao’s conjecture
over F (T ) for global function fields F . In this final section we shall check numerically
that (1.4) appears to satisfy a straightforward κ(u)(T )-analogue of Nagao’s conjecture, and
so the surprising behavior of fibral Mordell–Weil ranks in (1.4) (conditional on the parity
conjecture) does not seem to cast doubt on the standard reformulation of Nagao’s conjecture
in the case of pencils of elliptic curves over global function fields.
Nagao’s conjecture over Q(T ) involves the following data. Pick a non-constant elliptic
curve Eη over Q(T ), a prime p, and let Ep be the reduction modulo p. For all but finitely
many p, Ep is an elliptic curve over Fp (T ). Furthermore, the fiber Ep,s over each s ∈ P1 (Fp )
is an elliptic curve over Fp except for a set of s with size bounded uniformly in p. Set
1 X
(7.1)
Ap (Eη ) =
ap (Ep,s ),
p
1
s∈P (Fp )
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
33
where we only include points s for which the reduction Ep,s over Fp is an elliptic curve,
with ap denoting the Frobenius trace for such terms. Nagao conjectured that the rank of
Eη (Q(T )) can be computed as
1X
?
−Ap (Eη ) log p.
(7.2)
rank(Eη (Q(T ))) = lim
n→∞ n
p≤n
The existence of the limit is part of the conjecture, and bad p are omitted from the sum. (In
the definition (7.1), the omission of a uniformly bounded number of s for each p is irrelevant
1
for (7.2). Moreover, division by p rather than by p+1
) also has
P = #P (Fp√
√ no effect because
√
|ap | ≤ 2 p, by the Riemann hypothesis, and (1/n) p≤n (log p)/ p ∼ 2/ n = o(1).) Rosen
and Silverman [31] have shown the truth of (7.2) is closely related to a conjecture of Tate.
Nagao actually made two conjectures about formulas for ranks, one over Q [21, p. 213] and
the other over Q(T ) as above. The latter was inspired by the former.
The analogue of Nagao’s conjecture for an elliptic curve Eη over κ(u)(T ) ought to be the
following. For a place v on κ(u), with residue field Fv , set
X
1
av (Ev,s )
(7.3)
Av (Eη ) =
Nv
1
s∈P (Fv )
when Eη has good reduction over Fv (T ), and we only include s such that the reduction
Ev,s at s is an elliptic curve. Here Nv = #Fv and av is the Frobenius trace. Then, setting
q = #κ, the conjecture is
1 X
?
rank(E (κ(u)(T ))) = lim n
−Av (Eη ) deg v
n→∞ q
deg v=n
n X
= lim n
−Av (Eη ).
n→∞ q
deg v=n
(Allowing all v with (deg v)|n in the first sum adds a term O(n/q n/2 ) = o(1), so whether
we sum over those v satisfying deg v = n or (deg v)|n is irrelevant for the meaning of this
conjecture over κ(u).) As in the situation over Q, the Riemann hypothesis for elliptic curves
over finite fields ensures that this conjecture is unaffected by omitting terms in the Av (Eη )’s
corresponding to a set of points s ∈ P1 (Fv ) whose size is bounded uniformly in deg v.
We now consider this conjecture numerically for the curve in (1.4) where c = d = 1.
(Then h(T ) = T 2p + u in (3.2).) The generic rank is 1 by Theorem 1.1, so we expect
n
X
? q
(7.4)
−Av (Eη ) ∼ .
n
deg v=n
From the definition of Av in (7.3), clearly (7.4) is the same as
2n
X
X
? q
(7.5)
−av (Ev,s ) ∼
.
n
1
deg v=n s∈P (Fv )
Consider only n ≥ 2, since in degree 1 the specialization E∞ from (1.4) is not an elliptic
curve over κ(T ). Writing av in terms of the quadratic character on Fv , an equivalent
reformulation of (7.5) is
X
X
X x3 + h(s2 + u)x2 − h(s2 + u)3 x ? q 2n
(7.6)
∼
,
π
n
deg π=n s∈κ[u]/(π) x∈κ[u]/(π)
34
B. CONRAD, K. CONRAD, AND H. HELFGOTT
n Left side Right side
2
17
40.500
3
173
243.000
4
1186
1640.250
10788 11809.800
5
6
91816 88573.500
Table 4. (7.6) with q
Ratio
2.382
1.404
1.383
1.094
.964
=3
n Left side Right side
2
228
312.500
3
5430
5208.333
96802 97656.250
4
Table 5. (7.6) with q
Ratio
1.370
.959
1.008
=5
where π runs over monic irreducibles in κ[u] of degree n and ( π· ) is a Legendre symbol.
(Technically, we should be omitting from (7.6) the at most 8 values of s in each Fv such
that Ev,s is not an elliptic curve, but the inclusion of such terms in (7.6) contributes an
amount which has strictly smaller growth than q 2n /n, so there is nothing lost by allowing
such s for the sake of a cleaner summation.)
Tables 4 and 5 compare the two sides of (7.6) for q = 3 and q = 5. All decimal approximations are truncated after the third digit beyond the decimal point. This evidence
supports the truth of Nagao’s conjecture for (1.4). Thus, we do not find any evidence
that the formulation of Nagao’s conjecture for pencils over global function fields requires
unexpected restrictions. (We thank Aaron Silberstein for his assistance with the preceding
numerical calculations.)
Appendix A. Known results over Q
In the Introduction, we saw how to search for (conditional) examples of elevated rank
over Q(T ): assume the parity conjecture over Q and try to construct an elliptic curve over
Q(T ) that satisfies (1.2) for all but finitely many t ∈ P1 (Q). We wish to explain why
this sufficient strategy is essentially necessary if we also assume three additional standard
conjectures over Q. Moreover, we will see that if all of these conjectures are true then
there do not exist non-isotrivial examples of elevated rank over Q(T ). The three additional
conjectures we bring in are: the density conjecture, the squarefree-value conjecture, and
Chowla’s conjectures.
The density conjecture says that for any elliptic curve Eη over Q(T ), the rank of Et (Q)
equals rank(Eη (Q(T ))) or rank(Eη (Q(T ))) + 1 except for a set of t ∈ P1 (Q) with density 0,
as measured by height. Granting this and the parity conjecture, any example of elevated
rank over Q(T ) will satisfy (1.2) for all t outside of a set of density 0. Therefore, if Eη has
elevated rank then the average value of W (Et ), in the sense of the following definition, is
either 1 or −1.
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
35
Definition A.1. For any elliptic curve Eη over Q(T ), its average root number is
P
t∈P1 (Q),hQ (t)<N W (Et )
(A.1)
AvgQ W (Et ) := lim
N →∞ #{t ∈ P1 (Q), hQ (t) < N }
if this limit exists, where hQ is the standard logarithmic height function on P1 (Q) (defined
by the standard normalized collection of absolute values on Q). In the summation, the
finitely many t at which Et is non-smooth are dropped out.
The existence of the average root number is not evident a priori, and its value might
depend on the choice of coordinate on P1 . (The height hQ depends on the coordinate.) If
the average exists, then clearly −1 ≤ AvgQ W (Et ) ≤ 1. If we assume the parity and density
conjectures, then any example of elevated rank over Q(T ) must have average root number
1 or −1.
Remark A.2. For any elliptic curve E0 over Q, Rizzo [26] proved that the set of average
root numbers that unconditionally exist for quadratic twists of E0 over Q(T ) is dense in
the interval [−1, 1].
We now introduce the squarefree-value conjecture and Chowla’s conjectures; these lead
to a formula for AvgQ W (Et ) for any elliptic curve Eη over Q(T ). This formula turns out
never to equal 1 or −1 for non-isotrivial elliptic curves over Q(T ), thereby (conditionally)
ruling out the possibility of elevated rank for such elliptic curves.
The squarefree-value conjecture says that a polynomial over Z takes squarefree values
as often as is suggested by naive probabilistic heuristics. For example, if f (T ) ∈ Z[T ] is
squarefree, then the prediction is
#{1 ≤ n ≤ x : f (n) is squarefree} ∼ Cx
as x → ∞, where C = p 1 − cp /p2 with cp denoting the number of solutions to f (T ) = 0
in Z/(p2 ). (If 1 − cp /p2 = 0 for some p, then C = 0 and obviously f (n) is never squarefree.
Otherwise C is an absolutely convergent (positive) product.) We refer the reader to work
of Granville [9] for a more complete statement of this conjecture, including the variant
for homogeneous polynomials in two variables over Z. The squarefree-value conjecture is
known unconditionally for polynomials in Z[T ] with small degree, and Granville [9] deduced
the general case (all degrees) from the abc-conjecture. (Poonen [24] extended these results
to polynomials in any number of variables over Z, but only the cases treated by Granville
in one and two variables are related to the variation of root numbers in pencils of elliptic
curves over Q.)
Q
Remark A.3. Low-degree proved instances of the squarefree-value conjecture were used
in the study of ranks of elliptic curves over Q in [8], where families of quadratic twists were
considered.
The final conjecture we need over Q, due to Chowla [3, p. 96] in the one-variable case,
concerns the average behavior of the Liouville function on values of a polynomial. Recall
that Liouville’s function λ is the totally multiplicative function on Z defined by λ(±p) = −1
when p is prime, λ(±1) = 1, and λ(0) = 0.
The one-variable Chowla conjecture says that for any non-constant f (T ) in Z[T ], which
is not a perfect square up to sign, the sequence λ(f (n)) has average value 0 as n runs over
any arithmetic progression. That is, for any arithmetic progression a + bZ (with a ∈ Z and
36
B. CONRAD, K. CONRAD, AND H. HELFGOTT
b ∈ Z, b 6= 0), as N → ∞ we have
P
n∈(a+bZ)∩[0,N ] λ(f (n))
#((a + bZ) ∩ [0, N ])
→ 0.
(Clearly, with a linear change of variables, we can state this as a conjecture over all f using
only a = 0 and b = 1. We prefer the above superficially more general form because it
matches the two-variable conjecture more closely.)
The two-variable Chowla conjecture says that for any non-constant homogeneous f in
Z[U, V ] which is not a perfect square up to sign, the sequence λ(f (m, n)) has average value
0 as (m, n) runs over lattice points in any sector of the plane with vertex at the origin.
More precisely, for any coset L ⊆ Z2 of an arbitrary sublattice of Z2 , and any open sector
S ⊆ R2 with positive angular measure and vertex at the origin,
P
(m,n)∈S∩L∩[−N,N ]2 λ(f (m, n))
→0
(A.2)
#(S ∩ L ∩ [−N, N ]2 )
as N → ∞. If the condition (m, n) = 1 is imposed on the terms in the sum in (A.2), then
the resulting general conjecture is logically equivalent to the general conjecture (A.2).
In [12] and [13], the squarefree-value conjecture and the two-variable Chowla conjecture
are used to derive (conditional) formulas for AvgQ W (Et ) for any Eη/Q(T ) . The analysis falls
into two cases:
Case 1: The minimal regular proper model E → P1Q has no nodal geometric fiber. (That is,
Eη has no point of multiplicative reduction on P1Q .)
Case 2: The fibration E → P1Q has a nodal geometric fiber.
Consider a non-isotrivial Eη/Q(T ) in Case 1. Let Mt denote the finite set of primes p ∈ Z
such that Et has multiplicative reduction at p. The collection {Mt }t∈P1 (Q) is restricted as t
varies, in the following sense. Assuming the square-free value conjecture, we have that, for
any small ε > 0, there is a finite set of prime numbers Sε such that the set of t ∈ P1 (Q) with
Mt ⊆ Sε has height density ≥ 1 − ε. That is, roughly speaking, “most” fibers have their
primes of multiplicative reduction lying in a common finite set. (This remark is implicit in
[20, Lemma 2.1].) Moreover, for such t the bad primes for Et/Q outside of Sε are the prime
factors of values of certain irreducible primitive polynomials over Z that correspond to the
points of additive reduction for Eη on P1Q . (In particular, these primitive polynomials are
independent of t and ε.) For the study of average root numbers of elliptic curves over Q, the
essential difference between additive and multiplicative reduction is the simpler statistical
variation for local root numbers in the additive case. (See the formulas in Theorem 3.1.)
Assuming the squarefree-value conjecture, for “most” t the set of bad primes for Et outside
of Sε can be controlled, and a formula
Y
(A.3)
AvgQ W (Et ) = C∞
Cp
p
is thereby obtained,
where C∞ is an algebraic number in R, each Cp is a non-zero rational
Q
number, and p Cp is an absolutely convergent (non-zero) product.
Here are two examples that illustrate (A.3) (not elevated rank) for non-isotrivial elliptic
curves in Case 1.
Example A.4. An example of Washington [37, §3] over Q(T ) is
(A.4)
Eη : y 2 = x3 + T x2 − (T + 3)x + 1.
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
37
The point (0, 1) on this curve has infinite order (use Theorem 2.4, as in the proof of Corollary
2.6). Since E is a rational surface, it is not difficult to prove (using either analytic methods
of Rosen–Silverman or a reduction to positive characteristic and algebraic methods of Artin–
Tate) that Eη (K(T )) has rank 1 for every number field K.
In [27], Rizzo shows W (Et ) = −1 for every t ∈ Z. However, W (Et ) = 1 for many nonintegral t ∈ Q, such as (using PARI) t = −1/2, 1/3, and 3/2. An application of one of
the proved instances of the squarefree-value conjecture in low degree shows that (A.3) is
unconditionally true for Eη in (A.4), and a computation yields C∞ = 0. Therefore, in this
example, AvgQ W (Et ) = 0 unconditionally.
Example A.5. Let f (T ) = −5 − 2T 2 and g(T ) = 2 + 5T 2 . Consider
Eη : y 2 = x3 + a(T )x + b(T )
over Q(T ), where
54(f 3 + g 3 )(f 3 − g 3 )3
.
2
Low-degree proved instances of the squarefree-value conjecture imply that the conditional
formula (A.3) is true for this Eη . This leads to the explicit formula
Y ap
1
AvgQ W (Et ) = ·
1−
= 0.1527 . . . ,
6
(p + 1)2
a(T ) = −27f g(f 3 − g 3 )2 , b(T ) = −
p6=2,3,7,19
where ap = 1 + χp (−1) + (1 + χ3 (p))(1 + χ19 (p)) and χ` is the mod ` Legendre symbol.
A closer analysis of the work that leads to (A.3) in Case 1 shows that if the squarefreevalue conjecture is assumed then AvgQ W (Et ) cannot equal 1 or −1 in Case 1 when Eη is
non-isotrivial. Therefore, if the density conjecture, parity conjecture, and squarefree-value
conjecture are true, then in Case 1 there does not exist a non-isotrivial elliptic curve over
Q(T ) with elevated rank.
We turn now to Case 2, so E → P1Q has a nodal geometric fiber. Such E must be nonisotrivial. The reasoning in Case 1 breaks down, since there do not exist sets of t with height
density arbitrarily close to 1 such that the Et /Q ’s have multiplicative reduction in a common
finite set of primes. Now there is a non-constant homogeneous two-variable polynomial
fE ∈ Z[U, V ], which is not a square, such that as t ∈ P1 (Q) varies with Et/Q smooth, the
variation of the product of the local root numbers of Et/Q at the places of multiplicative
reduction is governed by the variation of λ(fE (m, n)) where m/n is the reduced form of t. A
similar phenomenon happens in our function field examples, using the Liouville function on
κ[u] that assigns value −1 to irreducibles and extends to all of κ[u] by total multiplicativity;
see Remark 3.3.) If we assume Chowla’s two-variable conjecture for fE , then the variation
of λ(fE (m, n)) as t = m/n varies can be controlled. Using this, in [12, §1.7] it is shown that
if the squarefree-value conjecture is also assumed, then AvgQ W (Et ) exists and equals 0; in
particular, this average does not equal 1 or −1. Thus, the parity, density, squarefree-value,
and Chowla conjectures predict that no elliptic curve in Case 2 has elevated rank.
Our discussion of AvgQ W (Et ) has shown that if we assume the squarefree-value conjecture
and Chowla’s two-variable conjecture then this average cannot equal 1 or −1 if Eη is nonisotrivial. If we accept the parity conjecture and the density conjecture, then any example
of elevated rank over Q(T ) has AvgQ W (Et ) = 1 or −1. Therefore, if all four conjectures
are true then all examples of elevated rank over Q(T ) must be isotrivial.
38
B. CONRAD, K. CONRAD, AND H. HELFGOTT
Appendix B. The surprise in characteristic p
We now replace Q with F = κ(u) and replace Z with κ[u], where κ is any finite field.
For the moment, κ may have characteristic 2. Granting the parity conjecture over F , no
new ideas should be required to construct isotrivial examples of elevated rank over F (T )
analogous to the examples of Cassels–Schinzel and Rohrlich. (The case of characteristic 2
is presumably more delicate.) We want to explain why it is reasonable to expect a priori
that non-isotrivial examples of elevated rank might exist over F (T ), despite the conclusions
over Q(T ) in Appendix A.
The squarefree-value conjecture for multivariable polynomials over κ[u], for κ with any
characteristic, was proved by Ramsay [25] in the separable case for one variable, and was
proved by Poonen [24] in general. Thus, provided that char(F ) 6= 2, 3 (to avoid problems
with wild ramification at arbitrarily many places), the methods used over Q(T ) can be
adapted to prove an unconditional formula akin to (A.3) in the analogue of Case 1 in
Appendix A. (This is the case of elliptic curves Eη/F (T ) such that E → P1F does not
have any nodal geometric fibers.) However, to adapt the Q(T )-methods to prove that
AvgF W (Et ) is strictly between 1 and −1 for a non-isotrivial E → P1F without nodal fibers,
we need to impose a restriction that is always satisfied in characteristic 0: the points in
the (non-empty) support of the conductor of Eη on P1F are étale over F . We expect that
if this étale restriction on the support of the conductor is dropped, then there should be
non-isotrivial examples without nodal geometric fibers such that the average root number
is 1 or −1. Moreover, in all positive characteristics there should exist such examples that
also have elevated rank (granting the parity conjecture).
Let us now turn to the analogue of Case 2 from Appendix A, so E → P1F has some nodal
geometric fibers. The study of such elliptic fibrations in characteristic 0 uses Chowla’s
conjectures over Z, as we saw in Appendix A. However, there are counterexamples to the
κ[u]-analogues of Chowla’s conjectures. In [4], it is shown that counterexamples to Chowla’s
one-variable conjecture are a common (but not “generic”) phenomenon. For example, elementary (but non-obvious) methods show that for any finite field κ with arbitrary characteristic p, f (T ) = T 4p +u ∈ κ[u][T ] violates the one-variable Chowla conjecture: λ(f (g)) = 1
for every g ∈ κ[u] with g 6∈ κ. Similarly, in the sense of Chowla’s two-variable conjecture,
the homogeneous polynomial
aX 4p + buY 4p ∈ κ[u][X, Y ]
with a, b ∈ κ× has rather non-random λ-values:
(
−1 if deg g1 ≤ deg g2 ,
4p
4p
(B.1)
λ(ag1 + bug2 ) =
1
if deg g1 > deg g2 ,
for any g1 , g2 ∈ κ[u] not both zero, using the convention deg 0 = −∞. (If p 6= 2 then (B.1) is
a special case of Lemma 3.5, replacing g1 and g2 in that lemma with their squares. We omit
the additional considerations that are required to verify (B.1) when p = 2.) In particular,
λ(ag14p + bug24p ) only depends on the sign of ord∞ (g1 /g2 ) = deg g2 − deg g1 . The proof of
Theorem 1.1 rests on a similar counterexample to Chowla’s two-variable conjecture, with
exponent 2p rather than 4p. (See (3.17).) The failure of Chowla’s conjecture in positive
characteristic was our initial clue to the possibility that elevated rank may occur in nonisotrivial families in the function field case.
ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC
39
References
[1] J. W. S. Cassels, A. Schinzel, “Selmer’s conjecture and families of elliptic curves,” Bull. London Math.
Soc. 14 (1982), 345–348.
[2] J. W. S. Cassels, “Second descents for elliptic curves,” J. Reine Angew. Math. 494 (1998), 101–127.
[3] S. Chowla, The Riemann hypothesis and Hilbert’s tenth problem, Gordon and Breach, New York, 1965.
[4] B. Conrad, K. Conrad, R. Gross, “Irreducible Specialization in Genus 0,” submitted to J. Reine Angew.
Math, 2004.
[5] B. Conrad, “Chow’s K/k-image and K/k-trace, and the Lang–Néron theorem,” unpublished notes
available at http://www.math.lsa.umich.edu/∼bdconrad.
[6] P. Deligne, “Les constantes des équations fonctionelles des fonctions L,” pp. 501–595 in Modular Functions of One Variable, II, Lecture Notes in Mathematics 349, Springer–Verlag, New York, 1973.
[7] G. Faltings, “Finiteness theorems for abelian varieties over number fields,” Ch. II in Arithmetic geometry
(Cornell/Silverman, ed.), Springer-Verlag, 1986.
[8] F. Gouvea, B. Mazur, “The square-free sieve and the rank of elliptic curves,” J. Amer. Math. Soc. 4
(1991), 1–23.
[9] A. Granville, “ABC allows us to count squarefrees,” Internat. Math. Res. Notices no. 19 (1998), 991–
1009.
[10] A. Grothendieck, “Eléments de géométrie algébrique,” Publ. Math. IHES, 4, 8, 11, 17, 20, 24, 28,
32 (1961-7).
[11] A. Grothendieck, “Séminaire de géométrie algébrique 3”, Springer Lecture Notes 151, Springer-Verlag,
New York, 1963.
[12] H. A. Helfgott, “Root numbers and the parity problem,” Ph.D. thesis, Princeton University,
http://www.arxiv.org/abs/math.NT/0305435.
[13] H. A. Helfgott, “On the behaviour of root numbers in families of elliptic curves,” submitted.
[14] A. J. de Jong, N. M. Katz, “Monodromy and the Tate conjecture: Picard numbers and Mordell–Weil
ranks in families,” Israel J. Math. 120 (2000), 47–79.
[15] K. Kato, F. Trihan, “On the conjectures of Birch and Swinnerton-Dyer in characteristic p > 0,” Invent.
Math. 153 (2003), 537–592.
[16] S. Kochen, “Integer-valued rational functions over the p-adic numbers: A p-adic analogue of the theory
of real fields,” in 1969 Number Theory (Proc. Sympos. Pure Math., Vol. XII) 57–73, Amer. Math. Soc.
Providence, RI, 1969.
[17] S. Lang, A. Néron, “Rational points on abelian varieties over function fields,” American J. Math. 81
(1959), 95–118.
[18] S. Lang, Abelian varieties, Interscience, New York, 1959.
[19] Q. Liu, Algebraic geometry and arithmetic curves, Oxford University Graduate Texts in Mathematics
6, Oxford University Press, Oxford, 2002.
[20] E. Manduchi, ”Root numbers of fibers of elliptic surfaces”, Compositio Math. 99 (1995) no. 1, 33–58.
[21] K. Nagao, “Construction of high-rank elliptic curves,” Kobe J. Math. 11 (1994), 211–219.
[22] K. Nagao, “Q(T )-rank of elliptic curves and certain limit coming from the local points,” Manuscripta
Math. 92 (1997), 13–32.
[23] J. Nekovář, “On the parity of ranks of Selmer groups. II,” C. R. Acad. Sci. Paris Sér. I Math. 332
(2001), no. 2, 99–104.
[24] B. Poonen, “Squarefree values of multivariable polynomials,” Duke Math. J. 118 (2003), 353–373.
[25] K. Ramsay, “Squarefree values of polynomials in one variable over function fields,” International Math.
Res. Notices no. 4 (1992), 97–102.
[26] O. Rizzo, “Average root numbers in families of elliptic curves,” Proc. Amer. Math. Soc. 127 (1999),
1597–1603.
[27] O. Rizzo, “Average root numbers for a non-constant family of elliptic curves,” Compositio Math. 136
(2003), 1–23.
[28] D. E. Rohrlich, “Variation of the root number in families of elliptic curves,” Compositio Math. 87
(1993), 119–151.
[29] D. E. Rohrlich, “Elliptic curves and the Weil-Deligne group,” in Elliptic curves and related topics,
125–157, CRM Proc. Lecture Notes 4 Amer. Math Soc., Providence, RI, 1994.
[30] D.E. Rohrlich, “Galois theory, elliptic curves, and root numbers,” Compositio Math. 100 (1996), 311–
349.
40
B. CONRAD, K. CONRAD, AND H. HELFGOTT
[31] M. Rosen and J. H. Silverman, “On the rank of an elliptic surface,” Invent. Math. 133 (1998), 43–67.
[32] K. Rubin, A. Silverberg, “Ranks of elliptic curves,” Bull. Amer. Math. Sci. 39 (2002), 455–474.
[33] J. Silverman, “Heights and the specialization map for families of abelian varieties,” J. Reine Angew.
Math. 342 (1983), 197–211.
[34] J. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1986.
[35] R. G. Swan, “Factorization of polynomials over finite fields,” Pacific J. Math. 12 (1962), 1099–1106.
[36] J. Tate, “Number theoretic background,” pp. 3–26 in: Automorphic Forms, Representations, and Lfunctions, Proc. Symp. Pure Math. 33 Part 2, Amer. Math. Soc., Providence, 1979.
[37] L. Washington, “Class numbers of the simplest cubic fields,” Math. Comp. 48 (1987), pp. 371–384.
Department of Mathematics, Univ. of Michigan, Ann Arbor, MI 48109-1109, USA
E-mail address: bdconrad@umich.edu
Department of Mathematics, Univ. of Connecticut, Storrs, CT 06269-3009, USA
E-mail address: kconrad@math.uconn.edu
Département de Mathématiques et Statistique, Université de Montréal, Montréal, QC
H3C 3J7, Canada
E-mail address: helfgott@dms.umontreal.ca